diff --git a/tex/energy_transfer/src/index.tex b/tex/energy_transfer/src/index.tex index c4cbc35..a45e5d5 100644 --- a/tex/energy_transfer/src/index.tex +++ b/tex/energy_transfer/src/index.tex @@ -400,7 +400,7 @@ stochastic process. \label{sec:multibath} 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 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 diff --git a/tex/gaussian_model/src/index.tex b/tex/gaussian_model/src/index.tex index 4ee93da..39a7106 100644 --- a/tex/gaussian_model/src/index.tex +++ b/tex/gaussian_model/src/index.tex @@ -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}). \end{multline} -This concludes the calculation. A possible measure would be to write -\cref{eq:bathderiv_1} as a sum of exponentials and give explicit -expressions for the coefficients and exponents. This is not required -for now. Code implementing this can be found under +This concludes the calculation. A possible measure of simplification +would be to write \cref{eq:bathderiv_1} as a sum of exponentials and +give explicit expressions for the coefficients and exponents. This is +not required for now. Code implementing this can be found under \url{https://github.com/vale981/hopsflow}. \section{Two Oscillators, Two Baths}% @@ -526,7 +526,7 @@ of the two harmonic oscillators. We find \begin{equation} \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}}. \end{equation} diff --git a/tex/hiromacros.sty b/tex/hiromacros.sty new file mode 100644 index 0000000..2724def --- /dev/null +++ b/tex/hiromacros.sty @@ -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}}} diff --git a/tex/hirostyle.sty b/tex/hirostyle.sty index 5595dd1..5e68f31 100644 --- a/tex/hirostyle.sty +++ b/tex/hirostyle.sty @@ -79,99 +79,3 @@ labelformat=brace, position=top]{subcaption} %% Minus Sign for Matplotlib \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)}}} diff --git a/tex/hops_tweaks/src/_preamble.tex b/tex/hops_tweaks/src/_preamble.tex index 0bf7db7..eea5ba0 100644 --- a/tex/hops_tweaks/src/_preamble.tex +++ b/tex/hops_tweaks/src/_preamble.tex @@ -3,14 +3,13 @@ captions=nooneline,captions=tableabove,english,DIV=16,numbers=noenddot,final]{scrartcl} \usepackage{../../hirostyle} +\usepackage{../../hiromacros} \usepackage{stackengine} \synctex=1 \title{HOPS Tweaks} \author{Valentin Link, Kai Mueller, Valentin Boettcher} \date{\today} -\newcommand{\mat}[1]{\ensuremath{{\underline{\vb{#1}}}}} -\def\kmat{{\mat{k}}} \begin{document} \maketitle \tableofcontents diff --git a/tex/hops_tweaks/src/index.tex b/tex/hops_tweaks/src/index.tex index 5d3d551..dd96335 100644 --- a/tex/hops_tweaks/src/index.tex +++ b/tex/hops_tweaks/src/index.tex @@ -68,7 +68,7 @@ the form \end{equation} where \(H_\sys\) is the (possibly time dependent) system Hamiltonian, \(H_B\nth = ∑_λω_λ\nth a_λ^{(n),†}a_λ\nth\), -\(B_n=∑_{λ}L_n^† g_λ\nth a_λ\nth\) 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 situation where each bath couples with the system through exactly one 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 \begin{multline} \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 \\+ - 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}} ]. \end{multline} @@ -239,7 +239,7 @@ are bosonic Fock-states. Now \cref{eq:multihops} becomes \begin{equation} \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 + - 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}