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give the open systems a once-over (thx richard)

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@ -92,9 +92,9 @@ state. Writing it down
\label{eq:schroedinger}
\iu ∂_{t} \ket{ψ(t)} = H \ket{ψ(t)},
\end{equation}
we find that we need to specify a \emph{Hamiltonian} \(H\) that acts
on our system state which is an element of a Hilbert space of some
dimension \(N\). Throughout the work we set \(\hbar=c=1\).
we find that we need only to specify a \emph{Hamiltonian} \(H\) that
acts on our system state which is an element of a Hilbert space of
some dimension \(N\). Throughout the work we set \(\hbar=c=1\).
We call the time evolution generated by \cref{eq:schroedinger}
\emph{Unitary}, as it preserves the norm of a state and is reversible.
@ -110,7 +110,7 @@ For time independent Hamiltonians the Schr\"odinger equation describes
a closed system which constitutes, within the scope of the problem in
question, the whole universe. In general, it is very hard to find a
closed expression for \cref{eq:time_evo_op}, except for very special
cases. Either one takes to approximations or one applies numerical
cases. Either we recourse to approximations or we apply numerical
methods to solve \cref{eq:schroedinger}.
When the Hilbert space dimension is small, its numerical solution is
@ -122,8 +122,13 @@ neglected when describing the descent of a space reentry capsule we
would arrive at fatally wrong results. Similarly, modern applications
of quantum physics deal with systems that undergo quantum evolution
under conditions that are not consistent with an isolated
system. Specifically in quantum computing~\cite{Gill2022Jan} the
effect of environmental interactions poses a major problem.
system.
Specifically in quantum computing~\cite{Gill2022Jan} the effect of
environmental interactions poses a major problem. Also, in quantum
thermodynamics, we are often concerned with irreversible dynamics that
converge to a steady state or a limit cycle. This is only possible, if
the bath possesses infininite degrees of freedom \cite{Breuer2002Jun}.
As a classical example, stokes drag models the influence of a viscous
fluid on spherical objects and can be implemented by adding a velocity
@ -133,46 +138,56 @@ dependent term to the equation of motion of our object,
\ddot{x} = F - α \dot{x}.
\end{equation}
We still retain all information about the system, the particle, having
accounted effectively for an environment, the fluid.
accounted effectively for an environment, the fluid by introducing a
term that breaks energy conservation.
In quantum physics, we find that the situation is more complicated.
Writing down a Hamiltonian we have to account for both system and
environment in a composite Hilbert space \(\hilb=\hilb_{\sys}\otimes\hilb_{\bath}\)
environment in a composite Hilbert space
\(\hilb=\hilb_{\sys}\otimes\hilb_{\bath}\)
\begin{equation}
\label{eq:general_open}
H = H_{\sys} \otimes \id_{\bath} + \id_{\sys} \otimes H_{\bath} + H_{\inter},
\end{equation}
where \(\sys\) marks the system, \(\bath\) marks the environment (or
bath) and \(H_{\inter}\) models the environment.
bath) and \(H_{\inter}\) models their interaction.
Although the global state of system and environment may be pure,
entanglement of system and environment leads to the effect, that we
may know the system state only as a statistical mixture of states
called the \emph{reduced state}. No part of a composite system may be
in general be known as ``precicely'' as the whole.
in general be known as ``precicely'' as the whole. This is a major
departure from classical physics.
Starting from a possibly mixed global state \(ρ(t)\) we find, that to
find the dynamics of all observables \(O_{\sys}\) that only act on the
system Hilbert space it is sufficient to know
compute the dynamics of all observables \(O_{\sys}\) which only act on
the system Hilbert space it is sufficient to know
\(ρ_{\sys}(t)=\tr_{\bath}[ρ(t)]\), the reduced system state.
The partial trace \(\tr_{\bath}\) averages over all bath degrees of
freedom and removes them from explicit consideration. This is a most
freedom and removes them from explicit consideration. This is a very
useful device, as the environment usually has a Hilbert space
dimension that is much too large for practical
calculations. Especially in numerically this fact is important, as
even an environment consisting of \(50\) two level systems would
consume \(128\) tebibyte of memory when stored as double precision
calculations. Especially in numerics this fact is important, as even
an environment consisting of \(40\) two level systems would consume
\(16\) tebibytes of memory when stored as double precision complex
floating point numbers.
We will discover in \cref{chap:flow} that although it is impractical
to work with the bath state directly, it is still possible gain access
to bath-global observables such as the change in the expectation value
of the bath energy.
Under certain assumptions, most importantly that of weak coupling
\(\ev{H_{\inter}}\approx 0\), a pertubative treatment of the
environment yields an evolution equation, called a \emph{master
equation}, that only contains the system state
\(ρ_{\sys}\)~\cite[p. 115 ff.]{Breuer2002Jun,Rivas2012}. This equation
often called Gorini–Kossakowski–Sudarshan–Lindblad equation, or GKSL
equation in short, leads to irreversible non unitary dynamics and has
the form
\(ρ_{\sys}\)~\cite[p. 115 ff.]{Breuer2002Jun,Rivas2012}, leading to
irreversible non unitary dynamics.
A special master equation often called
Gorini–Kossakowski–Sudarshan–Lindblad equation, or GKSL equation in
short, and has the form
\begin{equation}
\label{eq:gksl}
\dot{ρ}_{\sys} = -\iu \comm{H}{ρ_{\sys}} + \mathcal{D}[ρ_{\sys}] = \mathcal{L}[ρ_{\sys}],
@ -183,26 +198,26 @@ unitary contribution not necessarily equal to \(H_{\sys}\).
Integrating \cref{eq:gksl} leads to a map
\(ρ_{\sys}(t) = \mathcal{E}_{t}(ρ_{\sys}(0))\). The evolution
generated \cref{eq:gksl} is called Markovian, as
generated by \cref{eq:gksl} is called Markovian, as
\(\mathcal{E}_{t+s}= \mathcal{E}_{t}\circ\mathcal{E}_{s}\). More
fundamentally, this due to the fact that one at some point assumes,
that the bath has no ``memory''\footnote{This does not mean that the global
state has always the form \(ρ_{\sys}(t)\otimes
ρ_{\bath}(t)\)~\cite{Rivas2012}.}. Without getting into the details,
one may say that the characteristic time scales upon which correlation
functions of bath observables decay should be much smaller than the
time scales of the system.
fundamentally, this is due to the fact that at some point one assumes,
that the bath has no ``memory''\footnote{This does not mean that the
global state has always the form
\(ρ_{\sys}(t)\otimes ρ_{\bath}(t)\)~\cite{Rivas2012}.}. Without
getting into details, one may say that the characteristic time scales
upon which correlation functions of bath observables decay should be
much smaller than the time scales of the system.
If one endeavors to drop the assumptions of weak coupling and of
Markovian dynamics, the situation becomes more complicated. But when
introducing a concrete model of the bath we find that
\cref{eq:schroedinger} can be recast into a form that allows for an
exact numerical solution. The great advantage from the standpoint of
this thesis is, that although we solve for the reduced state
\(ρ_{\sys}\), we essentially calculate the unitary dynamics of system
and bath retain some information about the bath. This allows to
quantify the change in expected bath energy and also the expectation
value of the interaction Hamiltonian.
exact numerical solution. As alluded to above, the great advantage of
this exact approach is that although we solve for the reduced state
\(ρ_{\sys}\), we essentially compute the unitary dynamics of
system. Thus, we can additionally retain some information about the
bath which allows us to quantify the change in expected bath energy
and also the expectation value of the interaction Hamiltonian.
\Cref{sec:nmqsd_basics} will introduce the general model whose
solution will be made feasible with the introduction of the \emph{Non
@ -214,10 +229,6 @@ A more detailed account of both subjects can be found in
\cref{sec:multihops} as well as \cite{RichardDiss}.
The basics of the NMQSD will be briefly reviewed in
\cref{sec:nmqsd_basics} as will the basics of HOPS in
\cref{sec:hops_basics}.
\section{The Non Markovian Quantum State Diffusion (NMQSD)}
\label{sec:nmqsd_basics}