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Valentin Boettcher 2022-03-15 11:48:07 +01:00
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2
tex/energy_transfer/src/index.tex
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@ -400,7 +400,7 @@ stochastic process.
\label{sec:multibath} \label{sec:multibath}
For the models we consider in \fixme{citation,reference}, we have For the models we consider in \fixme{citation,reference}, we have
\([H_\inter^{(i)}, H_\inter^{(j)}] = 0\), where \(i,j\) are the bath \([H_\bath^{(i)}, H_\bath^{(j)}] = 0\), where \(i,j\) are the bath
indices. Therefore, we can apply the formalism of the previous indices. Therefore, we can apply the formalism of the previous
sections almost unchanged, by just taking care that all quantities sections almost unchanged, by just taking care that all quantities
involved in the expression of \(J_n=-\dv{\ev{H_B^{(n)}}}{t}\) refer to involved in the expression of \(J_n=-\dv{\ev{H_B^{(n)}}}{t}\) refer to

10
tex/gaussian_model/src/index.tex
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@ -373,10 +373,10 @@ and
∑_{m,n}\frac{A_nG_m}{C_n+W_m}\qty(1-\eu^{-(C_n+W_m)t}) - ∑_{m,n}\frac{A_nU_m}{C_n+Q_m}\qty(1-\eu^{-(C_n+Q_m)t}). ∑_{m,n}\frac{A_nG_m}{C_n+W_m}\qty(1-\eu^{-(C_n+W_m)t}) - ∑_{m,n}\frac{A_nU_m}{C_n+Q_m}\qty(1-\eu^{-(C_n+Q_m)t}).
\end{multline} \end{multline}
This concludes the calculation. A possible measure would be to write This concludes the calculation. A possible measure of simplification
\cref{eq:bathderiv_1} as a sum of exponentials and give explicit would be to write \cref{eq:bathderiv_1} as a sum of exponentials and
expressions for the coefficients and exponents. This is not required give explicit expressions for the coefficients and exponents. This is
for now. Code implementing this can be found under not required for now. Code implementing this can be found under
\url{https://github.com/vale981/hopsflow}. \url{https://github.com/vale981/hopsflow}.
\section{Two Oscillators, Two Baths}% \section{Two Oscillators, Two Baths}%
@ -526,7 +526,7 @@ of the two harmonic oscillators.
We find We find
\begin{equation} \begin{equation}
\label{eq:generalcorr} \label{eq:generalcorr}
C_{ij}(t, s) = G_{ik}(t)G_{jl}(s) C(0,0)_{ij} + C_{ij}(t, s) = G_{ik}(t)G_{jl}(s) C(0,0)_{kl} +
\underbrace{∫_0^t\dd{l}∫_0^s\dd{r}G_{ik}(t-l)G_{jl}(s-r) \ev{W_k(l)W_l(r)}}_{=Θ_{ij}}. \underbrace{∫_0^t\dd{l}∫_0^s\dd{r}G_{ik}(t-l)G_{jl}(s-r) \ev{W_k(l)W_l(r)}}_{=Θ_{ij}}.
\end{equation} \end{equation}

100
tex/hiromacros.sty Normal file
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@ -0,0 +1,100 @@
\ProvidesPackage{hiromacros}
% Macros
%% qqgg
\newcommand{\qqgg}[0]{q\bar{q}\rightarrow\gamma\gamma}
%% ppgg
\newcommand{\ppgg}[0]{pp\rightarrow\gamma\gamma}
%% Momenta and Polarization Vectors convenience
\DeclareMathOperator{\ps}{\slashed{p}}
\DeclareMathOperator{\pe}{\varepsilon}
\DeclareMathOperator{\pes}{\slashed{\pe}}
\DeclareMathOperator{\pse}{\varepsilon^{*}}
\DeclareMathOperator{\pses}{\slashed{\pe}^{*}}
%% Spinor convenience
\DeclareMathOperator{\us}{u}
\DeclareMathOperator{\usb}{\bar{u}}
\DeclareMathOperator{\vs}{v}
\DeclareMathOperator*{\vsb}{\overline{v}}
%% Center of Mass energy
\DeclareMathOperator{\ecm}{E_{\text{CM}}}
%% area hyperbolicus
\DeclareMathOperator{\artanh}{artanh}
\DeclareMathOperator{\arcosh}{arcosh}
%% Fast Slash
\let\sl\slashed
%% hermitian/complex conjugate
\DeclareMathOperator{\hc}{h.c.}
\DeclareMathOperator{\cc}{c.c.}
%% eulers number
\def\eu{\ensuremath{\mathrm{e}}}
%% Notes on Equations
\newcommand{\shorteqnote}[1]{ & & \text{\small\llap{#1}}}
%% Typewriter Macros
\newcommand{\sherpa}{\texttt{Sherpa}}
\newcommand{\rivet}{\texttt{Rivet}}
\newcommand{\vegas}{\texttt{VEGAS}}
\newcommand{\lhapdf}{\texttt{LHAPDF6}}
\newcommand{\scipy}{\texttt{scipy}}
%% Sherpa Versions
\newcommand{\oldsherpa}{\texttt{2.2.10}}
\newcommand{\newsherpa}{\texttt{3.0.0} (unreleased)}
%% Special Names
\newcommand{\lhc}{\emph{LHC}}
%% Expected Value and Variance
\newcommand{\EX}[1]{\operatorname{E}\qty[#1]}
\newcommand{\VAR}[1]{\operatorname{VAR}\qty[#1]}
%% Uppercase Rho
\newcommand{\Rho}{P}
%% Transverse Momentum
\newcommand{\pt}[0]{p_\mathrm{T}}
%% Sign Function
\DeclareMathOperator{\sign}{sgn}
%% Stages
\newcommand{\stone}{\texttt{LO}}
\newcommand{\sttwo}{\texttt{LO+PS}}
\newcommand{\stthree}{\texttt{LO+PS+pT}}
\newcommand{\stfour}{\texttt{LO+PS+pT+Hadr.}}
\newcommand{\stfive}{\texttt{LO+PS+pT+Hadr.+MI}}
%% GeV
\newcommand{\gev}[1]{\SI{#1}{\giga\electronvolt}}
\def\iu{\ensuremath{\mathrm{i}}}
\def\i{\iu}
\def\id{\ensuremath{\mathbb{1}}}
\def\RR{\ensuremath{\mathbb{R}}}
\def\CC{\ensuremath{\mathbb{C}}}
% fixme
\newcommand{\fixme}[1]{\textbf{\textcolor{red}{FIXME:~#1}}}
% HOPS/NMQSD
\def\sys{\ensuremath{\mathrm{S}}}
\def\bath{\ensuremath{\mathrm{B}}}
\def\inter{\ensuremath{\mathrm{I}}}
\def\nth{\ensuremath{^{(n)}}}
\newcommand{\mat}[1]{\ensuremath{{\underline{\vb{#1}}}}}
\def\kmat{{\mat{k}}}

96
tex/hirostyle.sty
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@ -79,99 +79,3 @@ labelformat=brace, position=top]{subcaption}
%% Minus Sign for Matplotlib %% Minus Sign for Matplotlib
\newunicodechar{−}{-} \newunicodechar{−}{-}
% Macros
%% qqgg
\newcommand{\qqgg}[0]{q\bar{q}\rightarrow\gamma\gamma}
%% ppgg
\newcommand{\ppgg}[0]{pp\rightarrow\gamma\gamma}
%% Momenta and Polarization Vectors convenience
\DeclareMathOperator{\ps}{\slashed{p}}
\DeclareMathOperator{\pe}{\varepsilon}
\DeclareMathOperator{\pes}{\slashed{\pe}}
\DeclareMathOperator{\pse}{\varepsilon^{*}}
\DeclareMathOperator{\pses}{\slashed{\pe}^{*}}
%% Spinor convenience
\DeclareMathOperator{\us}{u}
\DeclareMathOperator{\usb}{\bar{u}}
\DeclareMathOperator{\vs}{v}
\DeclareMathOperator*{\vsb}{\overline{v}}
%% Center of Mass energy
\DeclareMathOperator{\ecm}{E_{\text{CM}}}
%% area hyperbolicus
\DeclareMathOperator{\artanh}{artanh}
\DeclareMathOperator{\arcosh}{arcosh}
%% Fast Slash
\let\sl\slashed
%% hermitian/complex conjugate
\DeclareMathOperator{\hc}{h.c.}
\DeclareMathOperator{\cc}{c.c.}
%% eulers number
\def\eu{\ensuremath{\mathrm{e}}}
%% Notes on Equations
\newcommand{\shorteqnote}[1]{ & & \text{\small\llap{#1}}}
%% Typewriter Macros
\newcommand{\sherpa}{\texttt{Sherpa}}
\newcommand{\rivet}{\texttt{Rivet}}
\newcommand{\vegas}{\texttt{VEGAS}}
\newcommand{\lhapdf}{\texttt{LHAPDF6}}
\newcommand{\scipy}{\texttt{scipy}}
%% Sherpa Versions
\newcommand{\oldsherpa}{\texttt{2.2.10}}
\newcommand{\newsherpa}{\texttt{3.0.0} (unreleased)}
%% Special Names
\newcommand{\lhc}{\emph{LHC}}
%% Expected Value and Variance
\newcommand{\EX}[1]{\operatorname{E}\qty[#1]}
\newcommand{\VAR}[1]{\operatorname{VAR}\qty[#1]}
%% Uppercase Rho
\newcommand{\Rho}{P}
%% Transverse Momentum
\newcommand{\pt}[0]{p_\mathrm{T}}
%% Sign Function
\DeclareMathOperator{\sign}{sgn}
%% Stages
\newcommand{\stone}{\texttt{LO}}
\newcommand{\sttwo}{\texttt{LO+PS}}
\newcommand{\stthree}{\texttt{LO+PS+pT}}
\newcommand{\stfour}{\texttt{LO+PS+pT+Hadr.}}
\newcommand{\stfive}{\texttt{LO+PS+pT+Hadr.+MI}}
%% GeV
\newcommand{\gev}[1]{\SI{#1}{\giga\electronvolt}}
\def\iu{\ensuremath{\mathrm{i}}}
\def\i{\iu}
\def\id{\ensuremath{\mathbb{1}}}
\def\RR{\ensuremath{\mathbb{R}}}
\def\CC{\ensuremath{\mathbb{C}}}
% fixme
\newcommand{\fixme}[1]{\textbf{\textcolor{red}{FIXME:~#1}}}
% HOPS/NMQSD
\def\sys{\ensuremath{\mathrm{S}}}
\def\bath{\ensuremath{\mathrm{B}}}
\def\inter{\ensuremath{\mathrm{I}}}
\def\nth{\ensuremath{^{(n)}}}

3
tex/hops_tweaks/src/_preamble.tex
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@ -3,14 +3,13 @@
captions=nooneline,captions=tableabove,english,DIV=16,numbers=noenddot,final]{scrartcl} captions=nooneline,captions=tableabove,english,DIV=16,numbers=noenddot,final]{scrartcl}
\usepackage{../../hirostyle} \usepackage{../../hirostyle}
\usepackage{../../hiromacros}
\usepackage{stackengine} \usepackage{stackengine}
\synctex=1 \synctex=1
\title{HOPS Tweaks} \title{HOPS Tweaks}
\author{Valentin Link, Kai Mueller, Valentin Boettcher} \author{Valentin Link, Kai Mueller, Valentin Boettcher}
\date{\today} \date{\today}
\newcommand{\mat}[1]{\ensuremath{{\underline{\vb{#1}}}}}
\def\kmat{{\mat{k}}}
\begin{document} \begin{document}
\maketitle \maketitle
\tableofcontents \tableofcontents

10
tex/hops_tweaks/src/index.tex
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@ -68,7 +68,7 @@ the form
\end{equation} \end{equation}
where \(H_\sys\) is the (possibly time dependent) system Hamiltonian, where \(H_\sys\) is the (possibly time dependent) system Hamiltonian,
\(H_B\nth = ∑_λω_λ\nth a_λ^{(n),†}a_λ\nth\), \(H_B\nth = ∑_λω_λ\nth a_λ^{(n),†}a_λ\nth\),
\(B_n=∑_{λ}L_n^† g_λ\nth a_λ\nth\) and the \(L_n={(\vb{L})}_n\) are \(B_n=∑_{λ} g_λ\nth a_λ\nth\) and the \(L_n={(\vb{L})}_n\) are
arbitrary operators in the system Hilbert space. This models a arbitrary operators in the system Hilbert space. This models a
situation where each bath couples with the system through exactly one situation where each bath couples with the system through exactly one
spectral density and is therefore not fully general. spectral density and is therefore not fully general.
@ -218,9 +218,9 @@ Using
where \({\qty(\mat{e}_{n,μ})}_{ij}=δ_{ni}δ_{μj}\) we find after some algebra where \({\qty(\mat{e}_{n,μ})}_{ij}=δ_{ni}δ_{μj}\) we find after some algebra
\begin{multline} \begin{multline}
\label{eq:multihops} \label{eq:multihops}
\dot{ψ}^\kmat = \qty[-i H_\sys + \vb{L}\cdot\vb{η}^\ast - \dot{ψ}^\kmat = \qty[-\iu H_\sys + \vb{L}\cdot\vb{η}^\ast -
∑_{n=1}^N∑_{μ=1}^{M_n}\kmat_{n,μ}W\nth_μ]ψ^\kmat \\+ ∑_{n=1}^N∑_{μ=1}^{M_n}\kmat_{n,μ}W\nth_μ]ψ^\kmat \\+
i∑_{n=1}^N∑_{μ=1}^{M_n}\sqrt{G\nth_μ}\qty[\sqrt{\kmat_{n,μ}} L_nψ^{\kmat - \iu ∑_{n=1}^N∑_{μ=1}^{M_n}\sqrt{G\nth_μ}\qty[\sqrt{\kmat_{n,μ}} L_nψ^{\kmat -
\mat{e}_{n,μ}} + \sqrt{\qty(\kmat_{n,μ} + 1)} L^†_nψ^{\kmat + \mat{e}_{n,μ}} + \sqrt{\qty(\kmat_{n,μ} + 1)} L^†_nψ^{\kmat +
\mat{e}_{n,μ}} ]. \mat{e}_{n,μ}} ].
\end{multline} \end{multline}
@ -239,7 +239,7 @@ are bosonic Fock-states.
Now \cref{eq:multihops} becomes Now \cref{eq:multihops} becomes
\begin{equation} \begin{equation}
\label{eq:fockhops} \label{eq:fockhops}
∂_t\ket{Ψ} = \qty[-i H_\sys + \vb{L}\cdot\vb{η}^\ast - ∂_t\ket{Ψ} = \qty[-\iu H_\sys + \vb{L}\cdot\vb{η}^\ast -
∑_{n=1}^N∑_{μ=1}^{M_n}b_{n,μ}^\dag b_{n,μ} W\nth_μ + ∑_{n=1}^N∑_{μ=1}^{M_n}b_{n,μ}^\dag b_{n,μ} W\nth_μ +
i∑_{n=1}^N∑_{μ=1}^{M_n} \sqrt{G_{n,μ}} \qty(b^†_{n,μ}L_n + b_{n,μ}L^†_n)] \ket{Ψ}. \iu ∑_{n=1}^N∑_{μ=1}^{M_n} \sqrt{G_{n,μ}} \qty(b^†_{n,μ}L_n + b_{n,μ}L^†_n)] \ket{Ψ}.
\end{equation} \end{equation}