diff --git a/tex/hirostyle.sty b/tex/hirostyle.sty index f079ae1..5595dd1 100644 --- a/tex/hirostyle.sty +++ b/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)}}} diff --git a/tex/hops_tweaks/src/_preamble.tex b/tex/hops_tweaks/src/_preamble.tex index 782a7cd..0bf7db7 100644 --- a/tex/hops_tweaks/src/_preamble.tex +++ b/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 diff --git a/tex/hops_tweaks/src/index.tex b/tex/hops_tweaks/src/index.tex index c6107c5..7f56b6f 100644 --- a/tex/hops_tweaks/src/index.tex +++ b/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}