fix more typos

src/flow.tex

@@ -16,7 +16,7 @@ treated with the methods that will be developed as is shown in
 \cref{sec:general_obs}.
 
 The generalization to multiple baths in \cref{sec:multibath} and time
-depend Hamiltonians in \cref{sec:timedep} will present itself as
+dependent Hamiltonians in \cref{sec:timedep} will present itself as
 straight forward.
 
 \section{Bath Energy Change of a Zero Temperature Bath}%
@@ -161,7 +161,7 @@ Interestingly one finds that
 This expression is undesirable as it does not exist for all bath
 correlation functions\footnote{Only for BCFs that are smooth at
   \(τ=0\).} and expressions involving the process directly are alleged
-to converge slower, especially for shorter bath memories. This
+to converge more slowly, especially for shorter bath memories. This
 convergence problem is due to greater magnitude and shorter
 correlation time of the oscillations of \(\dot{η}_{t}^\ast\), as can
 be seen in \cref{fig:stocproc_comparison}.
@@ -169,7 +169,7 @@ be seen in \cref{fig:stocproc_comparison}.
   \centering
   \includegraphics{figs/analytic_comp/stocproc_comparison}
   \caption{\label{fig:stocproc_comparison} The imaginary part of ten
-    realizations the stochastic process \(η^\ast\) for an ohmic BCF
+    realizations of the stochastic process \(η^\ast\) for an ohmic BCF
     with different cutoff frequencies \(ω_{c}\). The process is much
     smoother and of less magnitude for smaller cutoffs. The difference
     between the cutoffs is even more severe for the derivative of the
@@ -352,7 +352,7 @@ and bath for a thermal initial condition as
   \rho(t) =
   \prod_\lambda\qty(∫\dd[2]{y_\lambda}
   \frac{\eu^{-\abs{y_\lambda}^2\bose_\lambda}}{\pi\bose_\lambda})
-  U(t)D(\vb{y})\bqty{\ketbra{\psi}\otimes\ketbra{0}}D(\vb{y})^† U(t)^†,
+  U(t)D(\vb{y})\bqty{\ketbra{\psi}\otimes\ketbra{0}}D^†(\vb{y}) U^†(t),
 \end{equation}
 where \(\bose_{λ}=\bose(βω_{λ})\). This is simply the
 Glauber-Sudarshan P representation of the bath state
@@ -364,7 +364,7 @@ The system state is then recovered through
   \rho_{\sys}(t) =
   \prod_\lambda\qty(∫\dd[2]{y_\lambda}
   \frac{\eu^{-\abs{y_\lambda}^2\bose_\lambda}}{\pi\bose_\lambda})
-  \tr_{\bath}\bqty{U(t)D(\vb{y})\bqty{\ketbra{\psi}\otimes\ketbra{0}}D(\vb{y})^† U(t)^†}.
+  \tr_{\bath}\bqty{U(t)D(\vb{y})\bqty{\ketbra{\psi}\otimes\ketbra{0}}D^†(\vb{y}) U^†(t)}.
 \end{equation}
 
 The usual step is now to insert \(\id =D(\vb{y})D^†(\vb{y})\) and
@@ -402,13 +402,13 @@ Remember that we want to calculate
                          \begin{aligned}
                            \prod_\lambda&\qty(∫\dd[2]{y_\lambda}
                                           \frac{\eu^{-\abs{y_\lambda}^2\bar{n}_\lambda}}{\pi\bar{n}_\lambda})\\
-                                        &\tr[L^†\dot{B}(t)
-                                          U(t)D(\vb{y})\ketbra{\psi}\otimes\ketbra{0}D(\vb{y})^† U(t)^†].
+                                        &\times\tr[L^†\dot{B}(t)
+                                          U(t)D(\vb{y})\ketbra{\psi}\otimes\ketbra{0}D^†(\vb{y}) U^†(t)].
                          \end{aligned}
   \end{aligned}
 \end{equation}
 To make a connection to the zero temperature results we again insert a
-\(\id\), but have to additionally commute \(D(\vb{y})^†\) past
+\(\id\), but have to additionally commute \(D^†(\vb{y})\) past
 \(\dot{B}(t)\). This leads to the expression
 \begin{equation}
   \label{eq:pureagain}
@@ -419,14 +419,14 @@ To make a connection to the zero temperature results we again insert a
                          \begin{aligned}
                            L^†(\dot{B}(t) + \dot{ξ}(t))
                            D^†(\vb{y}) &U(t)D(\vb{y})\ketbra{\psi}\\
-                                       &\otimes\ketbra{0}D^†(\vb{y})U(t)^†D(\vb{y})
+                                       &\otimes\ketbra{0}D^†(\vb{y})U^†(t)D(\vb{y})
                          \end{aligned}
                          ] \\
                        &=\prod_\lambda
                        \qty(∫\dd[2]{y_\lambda}
                          \frac{\eu^{-\abs{y_\lambda}^2\bar{n}_\lambda}}{\pi\bar{n}_\lambda})\\
                        &\qquad\times\tr[L^†\qty{\dot{B}(t) + \dot{ξ}(t)}
-                         \tilde{U}(t)\ketbra{\psi}\otimes\ketbra{0} \tilde{U}(t)^†],
+                         \tilde{U}(t)\ketbra{\psi}\otimes\ketbra{0} \tilde{U}^†(t)],
   \end{aligned}
 \end{equation}
 which indeed returns us to the zero temperature formalism with a transformed
@@ -465,7 +465,7 @@ Now,
 \begin{equation}
   \label{eq:hshcomm}
   [H_{\mathrm{sys}}^{\mathrm{shift}}, H_\inter] = ξ(t) [L^†, L]
-  B(t)^† + ξ^\ast(t) [L, L^†] B
+  B^†(t) + ξ^\ast(t) [L, L^†] B
 \end{equation}
 and therefore
 \begin{equation}
@@ -670,7 +670,7 @@ the form
 The corresponding version of~\cref{eq:f_ex_zero} would only explicitly
 depend on the zeroth order state and the stochastic processes. It has
 been observed that expressions involving the stochastic process
-directly tend to converge slower. However, this statement comes
+directly tend to converge more slowly. However, this statement comes
 without empirical proof and its verification may be left to future
 study. An explanation may be that the first hierarchy states fluctuate
 about their average dynamics whereas the stochastic process fluctuates

src/intro.tex

@@ -458,20 +458,20 @@ we can define the \(\vb{k}th\) hierarchy state
 
 The origin of the normalization chosen in \cref{eq:d_op_hier} is the
 desire to give all hierarchy states the same unit and to formulate the
-final hops equations unified into one equation in an enlarged Hilbert
+final HOPS equations unified into one equation in an enlarged Hilbert
 space~\cite{Gao2021Sep}. We refrain from going into details here and
 refer to \cref{sec:multihops} instead.
 
 For this state the following equation of motion can be
 derived~\cite{Suess2014Oct,Hartmann2021Aug}
-\begin{equation}
+\begin{multline}
   \label{eq:singlehops}\tag{HOPS}
   \ket{\dot{ψ}^{\vb{k}}} = \qty[-\iu H_\sys + \vb{L}\cdot\vb{η}^\ast -
-  ∑_{μ=1}^{M}k_{μ}W_μ]\ket{ψ^{\vb{k}}} +
-  \iu ∑_{μ=1}^{M}\sqrt{G_μ}\qty[\sqrt{k_{μ}}  L \ket{ψ^{\vb{k} -
+  ∑_{μ=1}^{M}k_{μ}W_μ]\ket{ψ^{\vb{k}}} \\
+  + \iu ∑_{μ=1}^{M}\sqrt{G_μ}\qty[\sqrt{k_{μ}}  L \ket{ψ^{\vb{k} -
     \vb{e}_{μ}}} + \sqrt{\qty(k_{μ} + 1)}  L^† \ket{ψ^{\vb{k} +
     \vb{e}_{μ}}} ],
-\end{equation}
+\end{multline}
 where \(\vb{k}=(k_{1}, k_{2}, \ldots, k_{M})\) with \(k_{μ}\geq 0\) is
 a multi index and \(\pqty{\vb{e}_{μ}}_{ν} = δ_{μ,ν}\). The term
 \({\vb{k} - \vb{e}_{μ}}\) is evaluated only if \(k_{μ}\geq 1\). The
@@ -529,7 +529,7 @@ The main code repository for this work can be found under
 \url{https://github.com/vale981/master-thesis}.
 
 The directory \path{python/energy_flow_proper} contains several
-project subdirectory with literate programming notebooks in the
+project subdirectories with literate programming notebooks in the
 \texttt{org} format~\cite{EricSchulte2022Sep}. A detailed listing
 linking subprojects to chapters can be found in
 \cref{tab:code_structure}.
@@ -564,7 +564,7 @@ merged into the original repository
 The extensive and well tested existing HOPS code of the Theoretical
 Quantum Optics group\footnote{Available upon reasonable request.} was
 created by Richard Hartmann. Some improvements have been made in the
-course of this work this work.  Documentation, type hints and unit
+course of this work.  Documentation, type hints and unit
 tests have been introduced. The parallelization mechanism was
 overhauled and now uses the \href{https://www.ray.io}{\texttt{Ray}}
 library. The structure of the code was further modularized, allowing

src/num_results.tex

@@ -11,7 +11,7 @@ verify the results of \cref{chap:flow} in
 \cref{sec:hopsvsanalyt}. Excellent consistency of the analytical and
 numeric solutions for the QBM models will be demonstrated.  A common
 feature of the short time behaviour of the bath energy flow that is
-visible in all simulations will be discussed and explained in
+visible in all simulations that will be discussed and explained in
 \cref{sec:pure_deph}.
 
 In the generic case where no analytic solution is known we
@@ -80,7 +80,7 @@ For the estimation of mean and standard deviation from trajectory
 data, Welford's online algorithm is employed to avoid catastrophic
 numerical cancellation~\cite{Welford1962Aug,Knuth1997}.
 
-In all simulations discussed use an Ohmic spectral density
+In all simulations, we will use an Ohmic spectral density
 \begin{equation}
   \label{eq:ohmic_sd}
   J(ω)=η ω \eu^{-\frac{ω}{ω_{c}}}\quad (ω>0)
@@ -567,7 +567,7 @@ We solve the model with the Hamiltonian (Schr\"odinger picture)
   \label{eq:puredeph}
   H = L^†(t) B + L(t) B^† + H_\bath
 \end{equation}
-with \(L(t)=L(t)^†\), \([L(t), L(s)] = 0\;\forall t,s\) (so that the
+with \(L(t)=L^†(t)\), \([L(t), L(s)] = 0\;\forall t,s\) (so that the
 Heisenberg Hamiltonian matches \cref{eq:puredeph}) and \(B,\,H_\bath\)
 as in \cref{sec:nmqsd_basics}.
 
@@ -630,9 +630,9 @@ match the zero temperature case in which the bath has minimal energy
 in the initial state.
 
 A thermodynamically useful model should feature significant system
-dynamics which is always given, assuming that the coupling is no too
+dynamics which is always given, assuming that the coupling is not too
 strong. Non hermitian coupling may also have this effect, but in the
-literature most effective qubit models tend to favour Hermitian
+literature most effective two-level models tend to favour Hermitian
 couplings
 \cite{Aurell2019Apr,Hita-Perez2021Nov,Hita-Perez2021Aug,MacQuarrie2020Sep,Andersen2017Feb,Mezzacapo2014Jul}. For
 the spin-boson model, non Hermitian coupling is the result of the
@@ -793,11 +793,11 @@ be witnessed in the inset of \Cref{fig:stocproc_systematics}.
 
 Only in the simulation with precision \(\varsigma=10^{-6}\) (blue)
 however, the compatibility condition of \cref{sec:meth} is satisfied
-for the given trajectory count. This is due to the fact that multiple
-quantities are being obtained in different ways, namely system energy
-and the flow. The system energy can be calculated directly from the
-system state whereas the flow has to be integrated numerically to
-obtain the total bath energy change.
+for the given trajectory count.  The system energy can be calculated
+directly from the system state whereas the flow has to be integrated
+numerically to obtain the total bath energy change and depends
+directly on the first hierarchy states. The accuracy must be high for
+the results to be consistent.
 \begin{figure}[htp]
   \centering
   \includegraphics{figs/one_bath_syst/stocproc_systematics_bath_energy}
@@ -821,7 +821,7 @@ plotted in \cref{fig:stocproc_consistency_dev}. Only for the highest
 precision (blue) case good consistency is given continually. For lower
 precision (green, orange), the consistency fluctuates and only
 occasionally surpasses \(68\%\). Initially compatibility is being
-demonstrated (until about \(N=10^4\)) but a divergence diverge from
+demonstrated (until about \(N=10^4\)) but a divergence from
 the most precise result occurs. It is therefore important to consider
 the dependence of the compatibility on the sample count \(N\) to judge
 the veracity of the simulation results.
@@ -854,7 +854,7 @@ convergence as is also demonstrated in
 
 \subsection{Hierarchy Truncation}
 \label{sec:trunc}
-As the systematics of the truncation depth has already been studied
+As the systematics of the truncation depth have already been studied
 thoroughly in~\cite{RichardDiss,Hartmann2021Aug}, we will keep the
 discussion short.  We chose \(N=4.5 \cdot 10^5\) trajectories and an
 Ohmic BCF with \(α(0)=0.8\) and \(ω_c=2\). Again, a BCF expansion with