more hops tweaks texed

tex/hirostyle.sty

@@ -169,3 +169,9 @@ labelformat=brace, position=top]{subcaption}
 
 % 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)}}}

tex/hops_tweaks/src/_preamble.tex

@@ -8,6 +8,9 @@ captions=nooneline,captions=tableabove,english,DIV=16,numbers=noenddot,final]{sc
 \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

tex/hops_tweaks/src/index.tex

@@ -56,3 +56,190 @@ These choices lead to an altered HOPS equation
     \tilde{φ})}{\norm{\tilde{ψ}}^2}+\qty(1-\norm{\tilde{ψ}}^2)]\mqty(\tilde{ψ}\\
   \tilde{φ}) + \mqty(F(\tilde{ψ},\tilde{φ}) \\ G(\tilde{ψ},\tilde{φ})).
 \end{equation}
+
+\section{Multiple Baths}
+\label{sec:multibath}
+
+We generalize the NMQSD and HOPS to \(N\) baths for Hamiltonians of
+the form
+\begin{equation}
+  \label{eq:multimodel}
+  H = H_\sys + ∑_{n=1}^N \qty[H_\bath\nth + \qty(L_n^†B_n + \hc)],
+\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
+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.
+
+\subsection{NMQSD}
+\label{sec:nmqsd}
+
+Following the usual derivation of the NMQSD \cite{Diosi1998Mar}, we
+switch to an interaction picture with respect to the \(H_\bath\)
+leading to
+\begin{equation}
+  \label{eq:multimodelint}
+  H(t) = H_\sys + ∑_{n=1}^N \qty[L_n^†B_n(t) + \hc],
+\end{equation}
+with \(B_n=∑_{λ}L_n^† g_λ\nth a_λ\nth\eu^{-\iu ω_λ\nth t}\).
+
+We will discuss the zero temperature case. The finite temperature
+methods generalize straight forwardly to multiple baths.  Projecting
+on a Bargmann (unnormalized) coherent state basis
+\(\qty{\ket{\vb{z}^{(1)},\vb{z}^{(2)},\ldots}=
+  \ket{\underline{\vb{z}}}}\) of the baths
+\begin{equation}
+  \label{eq:projected}
+  \ket{ψ(t)} = ∫∏_{n=1}^N{\qty(\frac{\dd{\vb{z}\nth}}{π^{N_n}}\eu^{-\abs{\vb{z}}^2})}\ket{ψ(t,\underline{\vb{z}}^\ast)}\ket{\underline{\vb{z}}},
+\end{equation}
+where \(N_n\) are the number of oscillators in each bath.
+
+
+We define
+\begin{equation}
+  \label{eq:processes}
+  η^\ast_n(t) = {\qty(\vb{η}^\ast_t)}_n= -\iu ∑_λg_λ^{(n),\ast} z_λ^{(n),\ast}\eu^{\iu ω_λ\nth t}
+\end{equation}
+and using
+\(\pdv{z_λ^{(n),\ast}}=∫\dd{s}\pdv{η^\ast_n(s)}{z_λ^{(n),\ast}}\fdv{η^\ast_n(s)}\)
+we arrive at
+\begin{equation}
+  \label{eq:multinmqsd}
+  ∂_tψ_t(\vb{η}^\ast_t) = -\iu H ψ_t(\vb{η}^\ast_t) +
+  \vb{L}\cdot\vb{η}^\ast_tψ_t(\vb{η}^\ast_t) - ∑_{n=1}^N L_n^†∫_0^t\dd{s}α_n(t-s)\fdv{ψ_t(\vb{η}^\ast_t)}{η^\ast_n(s)},
+\end{equation}
+where \(α_n(t-s)= {\qty(\vb{α}(t-s))}_n=∑_λ\abs{g_λ\nth}^2\eu^{-\iu ω_λ\nth(t-s)}\) are the
+zero temperature bath correlation functions. The equation
+\cref{eq:multinmqsd} becomes the NMQSD by reinterpreting the
+\(\vb{z}\nth\) as normal distributed complex random variables by
+virtue of monte-carlo integration of \cref{eq:projected}. The
+\(η^\ast_n(t)\) become homogeneous gaussian stochastic processes
+defined through
+\begin{equation}
+  \label{eq:processescorr}
+  \begin{aligned}
+      \mathcal{M}(η^\ast_n(t)) &=0, & \mathcal{M}(η_n(t)η_m(s)) &= 0,
+      & \mathcal{M}(η_n(t)η_m(s)^\ast) &= δ_{nm}α_n(t-s).
+  \end{aligned}
+\end{equation}
+
+\subsection{Nonlinear NMQSD}
+\label{sec:nonlin}
+
+For the derivation of the lonlinear theory, the characteristic
+trajectories of the partial differential equation of motion of
+the Husimi-function
+\begin{equation}
+  \label{eq:husimi}
+  Q_t(\underline{\vb{z}}, \underline{\vb{z}}^\ast) =
+  \frac{\eu^{-\abs{{\underline{\vb{z}}}}^2}}{π^{∑_n N_n}}
+  \braket{ψ(t, {\underline{\vb{z}}})}{ψ(t, {\underline{\vb{z}}}^\ast)}
+\end{equation}
+have to be determined.
+
+Using \(∂_{\underline{\vb{z}}}\ket{ψ(t, {\underline{\vb{z}}}^\ast)} =
+0\) and \(∂_{\underline{\vb{z}}^\ast}\bra{ψ(t, {\underline{\vb{z}}})} =
+0\) because \(\ket{ψ(t, {\underline{\vb{z}}}^\ast)}\) is holomorphic
+we derive
+\begin{equation}
+  \label{eq:husimimotion}
+  ∂_tQ_t(\underline{\vb{z}}, \underline{\vb{z}}^\ast) = -i
+  ∑_{n=1}^N\qty[∂_{z_λ^{(n), \ast}}\eu^{-\iu ω_λ\nth
+    t}\ev{L^†_n}_tQ_t(\underline{\vb{z}}, \underline{\vb{z}}^\ast) - \cc],
+\end{equation}
+where \(\ev{L^†_n}_t = \mel{ψ(t, {\underline{\vb{z}}})}{L^†_n}{ψ(t,
+  {\underline{\vb{z}}}^\ast)} / \braket{ψ(t, {\underline{\vb{z}}})}{ψ(t, {\underline{\vb{z}}}^\ast)}\).
+
+The characteristics of \cref{eq:husimimotion} obey the equations of
+motion
+\begin{equation}
+  \label{eq:characteristics}
+  \dot{z}^{(n),\ast}_λ = \iu g_λ\nth \eu^{-\iu ω_λ\nth t} \ev{L^†_n}_t
+\end{equation}
+for the stochastic state labels.
+
+The microscopic dynamics can in-turn be gathered into a shift of the
+stochastic processes
+\begin{equation}
+  \label{eq:procshift}
+  \tilde{η}_n^\ast(t) = η_n^\ast(t) + ∫_0^t\dd{s}α_n^\ast(t-s)\ev{L^†_n}_s
+\end{equation}
+and we obtain the nonlinear NMQSD equation
+\begin{multline}
+  \label{eq:multinmqsdnonlin}
+  ∂_tψ_t(\tilde{\vb{η}}^\ast_t) = -\iu H ψ_t(\tilde{\vb{η}}^\ast_t) +
+  \vb{L}\cdot\tilde{\vb{η}}^\ast_tψ_t(\tilde{\vb{η}}^\ast_t) \\-
+  ∑_{n=1}^N
+  \qty(L_n^†-\ev{L^†_n}_t)∫_0^t\dd{s}α_n(t-s)\eval{\fdv{ψ_t(\tilde{\vb{η}}^\ast_t)}{η^\ast_n(s)}}_{\vb{η}^\ast(s)
+  = \vb{η}(\underline{\vb{z}}^\ast(t), s)}.
+\end{multline}
+
+The notation
+\({\vb{η}^\ast(s) = \vb{η}(\underline{\vb{z}}^\ast(t), s)}\) means
+that we replace the microscopic \(z_λ^{(n),\ast}\) in
+\cref{eq:processes} with the shifted ones obeying
+\cref{eq:characteristics} and evaluate the resulting function at \(s\).
+This awkward construction can be remedied by the convolutionless
+formulation. It plays no great role in the HOPS formalism.
+
+\subsection{Multi Bath HOPS in Fock-Space Formulation}
+\label{sec:multihops}
+
+Following the usual derivation~\cite{RichardDiss} (but with a
+different normalization) and using an exponential expansion of the
+BCFs \(α_n(τ)=∑_{\mu}^{M_n}=G_μ\nth\eu^{-W_μ\nth τ}\), we define
+\begin{equation}
+  \label{eq:dops}
+  D_μ\nth(t) \equiv ∫_0^t\dd{s}\eu^{-W_μ\nth (t-s)}\fdv{η^\ast_n(s)}
+\end{equation}
+and
+\begin{equation}
+  \label{eq:dops}
+  D^{\underline{\vb{k}}} \equiv
+  ∏_{n=1}^N∏_{μ=1}^{M_n}
+  {\sqrt{\frac{\underline{\vb{k}}_{n,μ}!}{\qty(G\nth_μ)^{\underline{\vb{k}}_{n,μ}}}}
+  \frac{1}{i^{\underline{\vb{k}}_{n,μ}}}}\qty(D_μ\nth)^{\underline{\vb{k}}_{n,μ}},
+\end{equation}
+as well as
+\begin{equation}
+  \label{eq:hierdef}
+  ψ^{\underline{\vb{k}}} \equiv D^{\underline{\vb{k}}}ψ.
+\end{equation}
+
+Using
+\begin{equation}
+  \label{eq:commrelation}
+  [D^\kmat(t),η_n^\ast(t)] =  \iu∑_{μ=1}^{M_n}
+  \sqrt{\kmat_{n,μ}G\nth_μ} D^{\kmat -
+    \mat{e}_{n,μ}}
+\end{equation}
+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 -
+  ∑_{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 -
+    \mat{e}_{n,μ}} + \sqrt{\qty(\kmat_{n,μ} + 1)}  L^†_nψ^{\kmat +
+    \mat{e}_{n,μ}} ].
+\end{multline}
+
+The HOPS equations \cref{eq:multihops} can also be rewritten in an
+especially appealing form \cite{Gao2021Sep} if we embed the hierarchy
+states into a larger Hilbert space using
+\begin{equation}
+  \label{eq:fockpsi}
+  \ket{Ψ} = \sum_\kmat\ket{\psi^\kmat}\otimes \ket{\kmat}
+\end{equation}
+where
+\(\ket{\kmat}=\bigotimes_{n=1}^N\bigotimes_{μ=1}^{N_n}\ket{\kmat_{n,μ}}\)
+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 -
+  ∑_{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}.
+\end{equation}