some introduction work

references.bib

@@ -1347,3 +1347,33 @@
   publisher =    {American Physical Society},
   doi =          {10.1103/PhysRevE.103.042145}
 }
+
+@article{Gill2022Jan,
+  author =       {Gill, Sukhpal Singh and Kumar, Adarsh and Singh,
+                  Harvinder and Singh, Manmeet and Kaur, Kamalpreet
+                  and Usman, Muhammad and Buyya, Rajkumar},
+  title =        {{Quantum computing: A taxonomy, systematic review
+                  and future directions}},
+  journal =      {Softw.: Pract. Exper.},
+  volume =       52,
+  number =       1,
+  pages =        {66--114},
+  year =         2022,
+  month =        jan,
+  issn =         {0038-0644},
+  publisher =    {John Wiley {\&} Sons, Ltd},
+  doi =          {10.1002/spe.3039}
+}
+
+@book{Caldeira2014Mar,
+  author =       {Caldeira, A. O.},
+  title =        {{An Introduction to Macroscopic Quantum Phenomena
+                  and Quantum Dissipation}},
+  journal =      {Cambridge Core},
+  year =         2014,
+  month =        mar,
+  isbn =         {978-0-52111375-5},
+  publisher =    {Cambridge University Press},
+  address =      {Cambridge, England, UK},
+  doi =          {10.1017/CBO9781139035439}
+}

src/intro.tex

@@ -33,7 +33,7 @@ definition of heat, as is expounded in
 
 As it turns out, the framework of the ``Non Markovian State
 Diffusion'' (NMQSD)~\cite{Diosi1998Mar}, which will be reviewed
-in~\cref{sec:quick_hops}, allows access to certain bath related
+in~\cref{sec:open_systems}, allows access to certain bath related
 observables such as the time derivative of the bath energy expectation
 value and the interaction Hamiltonian expectation value. The
 
@@ -53,19 +53,152 @@ auxiliary bosonic Fock space in \cref{sec:multihops}.
 
 \newpage
 
-\section{Open Systems}
+\section{Open Quantum Systems}
-\label{sec:quick_hops}
+\label{sec:open_systems}
-\begin{itemize}
+Quantum physics' most important equation, the Schr\"odinger equation,
-\item set the stage, introduce open systems (not just dump the general
+allows us to predict the future of a system knowing its initial
-  model on people as is done currently\ldots)
+state. Writing it down
-\item usually closed systems, open systems only weakly coupled or
+\begin{equation}
-  ``effectively'' through statistical physics
+  \label{eq:schroedinger}
-\item emphasize the unitary time development of the total system as
+  \iu ∂_{t} \ket{ψ(t)} = H \ket{ψ(t)},
-  ``exact'' approach
+\end{equation}
-\end{itemize}
+we find that we need to specify a \emph{Hamiltonian} \(H\) that acts
-The basic and most general model which forms the foundation of all
+on our system state which is an element of a Hilbert space of some
-matters discussed in this is a general quantum system \(H_\sys(t)\)
+dimension \(N\). Throughout the work we set \(\hbar=c=1\).
-coupled to \(N\) baths of harmonic oscillators
+
+We call the time evolution generated by \cref{eq:schroedinger}
+\emph{Unitary}, as it preserves the norm of a state and is reversible.
+Given any time independent Hamiltonian we may write down the time
+evolution operator to solve the Schr\"odinger equation
+\begin{equation}
+  \label{eq:time_evo_op}
+  U(t, t_{0})=\eu^{-\iu H (t-t_{0})},\; U(t, t_{0})^\dag U(t, t_{0}) =
+  \id,\; U\ket{ψ(t_{0})} = \ket{ψ(t)}.
+\end{equation}
+
+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
+methods to solve \cref{eq:schroedinger}.
+
+When the Hilbert space dimension is small, its numerical solution is
+straight forward. But in more realistic scenarios we may still be
+interested in a small system, but we cannot neglect the interaction of
+that system with a much larger environment sometimes consisting of
+infinite degrees of freedom. If the atmosphere of the earth would be
+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.
+
+As a classical example, stokes drag models the influence of a viscous
+fluid on spherical objects and can be implemented by adding a velocity
+dependent term to the equation of motion of our object,
+\begin{equation}
+  \label{eq:newton}
+  \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.
+
+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 \(\hilb=\hilb_{\sys}\otimes\hilb_{\bath}\)
+\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.
+
+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.
+
+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
+\(ρ_{\sys}(t)=\tr_{\bath}[ρ(t)]\), 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
+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
+floating point numbers.
+
+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
+\begin{equation}
+  \label{eq:gksl}
+  \dot{ρ}_{\sys} = -\iu \comm{H}{ρ_{\sys}} + \mathcal{D}[ρ_{\sys}] = \mathcal{L}[ρ_{\sys}],
+\end{equation}
+where \(\mathcal{D}\) is called the \emph{dissipator} which adds
+non-unitary dynamics to the von Neumann equation and \(H\) is a
+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
+\(\mathcal{E}_{t+s}= \mathcal{E}_{t}\circ\mathcal{E}_{s}\). 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.
+
+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.
+
+\Cref{sec:nmqsd_basics} will introduce the general model whose
+solution will be made feasible with the introduction of the \emph{Non
+  Markovian Quantum State Diffusion} (NMQSD). The numerical
+implementation of the NMQSD, the \emph{Hierarch of Pure States}
+(HOPS), will be the topic of \cref{sec:hops_basics}.
+
+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}
+\label{sec:nmqsd_basics}
+
+We will now introduce the fundamental form of the models that will be
+discussed in this thesis. This model has a wide applicability and many
+microscopic systems can be cast into its form
+\cite{Strunz2001Habil}\cite[chap. 2]{RichardDiss}, although there
+certainly exist limits of applicability~\cite{Caldeira2014Mar}.
+
+Consider a general quantum system \(H_\sys(t)\) coupled to \(N\) baths
+of harmonic oscillators\footnote{For instance, the electromagnetic field.}
 \begin{equation}
   \label{eq:generalmodel}
   H(t) = H_\sys(t) + ∑_{n=1}^N \qty[L_n^†(t)B_n + \hc] + ∑_{n=1}^NH_B\nth ,
@@ -73,36 +206,24 @@ coupled to \(N\) baths of harmonic oscillators
 with \(B_n=∑_{λ} g_λ\nth a_λ\nth\) and
 \(H_B\nth=∑_λω_λ\nth \qty(a_λ\nth)^\dag a_λ\nth\). The \(a_λ\) are
 bosonic annihilation operators and the \(L_n\) are arbitrary not
-necessarily hermitian operators system Hilbert space. Sometimes
+necessarily hermitian operators system Hilbert space.
-\cref{eq:generalmodel} is called the ``Standard Model of Open
-Systems''. Throughout the work we set \(\hbar=c=1\).
 
 Despite the simple structure of the baths, \cref{eq:generalmodel} is
-generally very hard to solve beyond weak coupling strengths and the
+generally very hard to solve beyond weak coupling strengths as has
-secular approximation~\cite{Rivas2012}. The ``Non Markovian Quantum
+been detailed in~\cref{sec:open_systems}. The ``Non Markovian Quantum
 State Diffusion'' (NMQSD)~\cite{Diosi1998Mar} approach allows to
 recast \cref{eq:generalmodel} into a stochastic differential equation
 in which the bath degrees of freedom are accounted for by Gaussian
-stochastic processes. This drastic reduction of the number of degrees
+stochastic processes.
-of freedom also leads to an efficient numerical method, the
-``Hierarchy of Pure States'' (HOPS).
 
-The basics of the NMQSD will be briefly reviewed in
+Here we only consider a single zero temperature bath initially in the
-\cref{sec:nmqsd_basics} as will the basics of HOPS in
+ground state \(\ket{0}\). For more complete and general account see
-\cref{sec:hops_basics}. A more detailed account may be found in
+\cite{RichardDiss,Strunz2001Habil,Diosi1998Mar,Hartmann2017Dec} and
-\cref{sec:multihops} as well as \cite{RichardDiss}.
+\cref{sec:hops_multibath}.
 
-\section{The Non Markovian Quantum State Diffusion}
+The total system-bath state may then be expanded in a Bargmann
-\label{sec:nmqsd_basics}
+coherent state basis~\cite{klauder1968fundamentals} with respect to
-\begin{itemize}
+the bath degrees of freedom
-\item after general intro: technical basics for \cref{chap:flow}
-\item problem: hilbert space too large
-\item solution: exploit structure of bath and recast problem
-\end{itemize}
-We begin by considering a single zero temperature bath in the ground
-state \(\ket{0}\). The total system-bath state may then be expanded in
-a Bargmann coherent state basis~\cite{klauder1968fundamentals} with
-respect to the bath degrees of freedom
 \begin{equation}
   \label{eq:projected_single}
   \ket{ψ(t)} = ∫{\frac{\dd{\vb{z}}}{π^{N}}\eu^{-\abs{\vb{z}}^2}}\ket{ψ(t,\vb{z})^\ast}\ket{\vb{z}},
@@ -114,9 +235,9 @@ After transforming \cref{eq:generalmodel} into the interaction picture
 with respect to \(H_B\) and using the properties of the coherent
 states (\(\mel{z_λ}{a_λ}{ψ}\rightarrow ∂_{z_λ^\ast}\braket{z_λ}{ψ}\),
 \(\mel{z_λ}{a_λ^\dag}{ψ}\rightarrow z_λ^\ast\braket{z_λ}{ψ}\)) we
-arrive at
+arrive at an equation for stochastic pure state trajectories
 \begin{equation}
-  \label{eq:nmqsd_single}
+  \label{eq:nmqsd_single}\tag{NMQSD}
   ∂_tψ_t(η^\ast_t) = -\iu H ψ_t(η^\ast_t) +
   L {η}^\ast_tψ_t({η}^\ast_t) - L^†∫_0^t\dd{s}α(t-s)\fdv{ψ_t({η}^\ast_t)}{η^\ast_s},
 \end{equation}
@@ -129,11 +250,18 @@ and \(η_t\) is a Gaussian stochastic process obeying
 \begin{equation}
   \label{eq:single_processescorr}
   \begin{aligned}
-      \mathcal{M}(η^\ast_t) &=0, & \mathcal{M}(η_tη_s) &= 0,
+      \mathcal{M}(η_t) &=0, & \mathcal{M}(η_tη_s) &= 0,
       & \mathcal{M}(η_tη_s^\ast) &= α(t-s).
   \end{aligned}
 \end{equation}
 
+The reduced system state may then be recovered by averaging over all
+trajectories
+\begin{equation}
+  \label{eq:recover_rho}
+  ρ_{\sys}(t) = \mathcal{M}_{η_{t}^\ast}\bqty{ψ_t(η_t)^\dag ψ_t(η^\ast_t)}.
+\end{equation}
+
 Note that the BCF \(α\) is usually defined as Fourier transform of the
 spectral density
 \begin{equation}
@@ -142,14 +270,110 @@ spectral density
 \end{equation}
 One then usually performs a continuum limit so that \(J(ω)\) becomes
 ``smeared out'' to a smooth function and \(α(τ)\) decays to zero for
-\(τ\rightarrow ∞\). This behavior leads to \cref{eq:nmqsd_single}
+\(τ\rightarrow ∞\).
+
+We have found that indeed we can treat an infinite environment with a
+stochastic differential equation in which only objects of system
+dimension appear. Note also, that we can treat explicit time
+dependence of \(L\) and \(H\) without alteration to
+\cref{eq:nmqsd_single}.
+
+
+The equation \cref{eq:nmqsd_single} does not preserve the norm of the
+state, leading to suboptimal convergence of \cref{eq:recover_rho}.
+To remedy this, we choose a co-moving shifted stochastic process
+\begin{equation}
+  \label{eq:shifted_proc}
+  \tilde{η}_{t}^\ast= η^\ast_{t} + ∫_{0}^{t}\dd{s} α^\ast(t-s) \ev{L^\ast}_{s},
+\end{equation}
+where
+\(\ev{L^\dag}_{t}=ψ(\tilde{η}_{t}^\ast)_{t}^\dag L^\dag
+ψ(\tilde{η}_{t}^\ast)_{t}\).
+
+This leads to the nonlinear NMQSD equation
+\begin{equation}
+  \label{eq:nmqsd_nonlin_single}
+  ∂_tψ_t(η^\ast_t) = -\iu H ψ_t(η^\ast_t) +
+  L {η}^\ast_tψ_t(\tilde{η}^\ast_t) - \pqty{L^† -\ev{L^\dag}_{t}}∫_0^t\dd{s}α(t-s)\fdv{ψ_t({η}^\ast_t)}{\tilde{η}^\ast_s}.
+\end{equation}
+There is a subtlety concerning the functional derivative that won't be
+discussed here, but in \cref{sec:nonlin}.
+Crucially, the system state is now recovered through
+\begin{equation}
+  \label{eq:recover_rho_nonlinear}
+  ρ_{\sys}(t) = \mathcal{M}_{η_{t}^\ast}\bqty{\frac{ψ_t(η_t)^\dag ψ_t(η^\ast_t)}{\norm{ψ_t(η_t)}^{2}}},
+\end{equation}
+so that all trajectories contribute with ``equal weight''.
+
 
 \section{The Hierarchy of Pure States}
 \label{sec:hops_basics}
-\begin{itemize}
+The equation \cref{eq:nmqsd_single} has removed the bath degrees of
-\item point out difficulties with the functional derivative
+freedom from explicit consideration, replacing them with a Gaussian
-\item goal: numerical method for general systems: mutli bath,
+stochastic process and a rather complicated term containing a memory
-  modulated etc
+integral and a functional derivative
-\item introduce HOPS briefly and refer to \cref{chap:hops_notes} for
+\begin{equation}
-  details
+  \label{eq:complicated_term}
-\end{itemize}
+  ∫_0^t\dd{s}α(t-s)\fdv{ψ_t({η}^\ast_t)}{η^\ast_s}.
+\end{equation}
+
+There exist analytical approaches to this
+term~\cite{Diosi1998Mar,Strunz2001Habil}, but we keep the approach as
+general as possible and instead choose a numerical avenue.
+
+They key is define away the complicated term containing the functional
+derivative as an auxiliary state. Expanding the BCF into exponentials
+\(α(τ)=∑_{μ}G_{μ=1}^{M}\eu^{-W_{μ}τ}\) and defining
+\begin{equation}
+  \label{eq:d_op_one}
+  D_{μ}(t)\equiv ∫_{0}^{t} G_{μ} \eu^{-W_{μ}(t-s)} \fdv{η^\ast_s},\; D^{\vb{k}}(t)\equiv Π_{μ=1}^{M}\sqrt{\frac{k_{μ}!}{G_{μ}^{k_{μ}}}}
+  \frac{1}{i^{k_{μ}}}\pqty{D_{μ}}^{k_{μ}}
+\end{equation}
+we can define the \(\vb{k}th\) hierarchy state
+\begin{equation}
+  \label{eq:d_op_hier}
+   ψ^{\vb{k}}\equiv D^{\vb{k}}ψ.
+\end{equation}
+
+For this state the following equation of motion can be derived
+\begin{equation}
+  \label{eq:singlehops}\tag{HOPS}
+  \dot{ψ}^{\vb{k}} = \qty[-\iu H_\sys + \vb{L}\cdot\vb{η}^\ast -
+  ∑_{μ=1}^{M}k_{μ}W_μ]ψ^{\vb{k}} +
+  \iu ∑_{μ=1}^{M}\sqrt{G_μ}\qty[\sqrt{k_{μ}}  L ψ^{\vb{k} -
+    \vb{e}_{μ}} + \sqrt{\qty(k_{μ} + 1)}  L^† ψ^{\vb{k} +
+    \vb{e}_{μ}} ],
+\end{equation}
+where \(\vb{k}=(k_{1}, k_{2}, \ldots, k_{M})\) with \(k_{μ}\geq 0\) is
+a multi index an \(\pqty{\vb{e}_{μ}}_{ν} = δ_{μ,ν}\). The term
+\({\vb{k} - \vb{e}_{μ}}\) is evaluated only if \(k_{μ}\geq 1\). We
+call \(\abs{\vb{k}}=∑_{μ}k_{μ}\) the hierarchy level of
+\(ψ^{\vb{k}}\). The state \(ψ\equiv ψ^{\vb{0}}\) corresponds to the
+trajectory obeying \cref{eq:nmqsd_single}.
+
+
+We call \cref{eq:singlehops} the \emph{Hierarchy of Pure States}
+because each hierarchy state couples only to the hierarchy states one
+level above and one level below. This is similar to the
+\emph{Hierarchical Equations of Motion} (HEOM) approach used
+in~\cite{Kato2016Dec}, but with the advantage of reducing the
+dimensionality from \(\dim{\hilb_{\sys}}^{2}\) to
+\(\dim{\hilb_{\sys}}\) by treating pure states instead of density
+matrices.
+
+By truncating the hierarchy we obtain from \cref{eq:singlehops} a
+linear differential equation that can be solved numerically. By
+choosing a suitable cutoff the method can be made arbitrarily
+exact.
+
+Note that the hierarchy states have no physical interpretation, but
+can be thought of as the ``memory'' of the system. We will find in
+\cref{chap:flow} that they can be of use beyond the mere calculation
+of \(ψ=ψ^{\vb{0}}\).
+
+The nonlinear NMQSD \cref{eq:nmqsd_nonlin_single} can be accommodated
+in much the same way, except for the replacements
+\(L^\dag\rightarrow \pqty{L^\dag-\ev{L^\dag}_{t}}\) and
+\(η\rightarrow \tilde{η}\) in \cref{eq:singlehops}. Throughout this
+work, the nonlinear method is being used, as it offers much superior
+convergence.