nonlinear flow cleanup

references.bib

@@ -69,7 +69,7 @@
 }
 
 @article{Diosi1995Jan,
-  author =       {Di{\ifmmode\acute{o}\else\'{o}\fi}si, Lajos and
+  author =       {Diósi, Lajos and
                   Gisin, Nicolas and Halliwell, Jonathan and Percival,
                   Ian C.},
   title =        {{Decoherent Histories and Quantum State Diffusion}},

src/flow.tex

@@ -180,61 +180,55 @@ states.
 
 \section{Generalization to the Nonlinear Theory}
 \label{sec:nonlin_flow}
-\begin{itemize}
-\item until now: not much use for stronger coupling
-\item nonlinear method way better convergence
-\item we show that results be generically generalized to the nonlinear
-  theory by ``just normalizing''
-\end{itemize}
 Due to the inferior convergence of the linear method for stronger
 coupling \cite{Suess2014Oct}, the results of \cref{sec:flow_lin} are
 not yet of much practical use. We therefore turn to the nonlinear
 NMQSD which preserves the norm of the stochastic trajectory and
 ammounts to performing importance sampling at each point in time.
 
-In the spirit of the usual derivation of the nonlinear NMQSD we write
+Following the usual derivation of the nonlinear NMQSD we write
 \begin{equation}
   \label{eq:newb}
   \begin{aligned}
-  \ev{L^†\dot{B}(t)} &= ∫ \frac{\dd[2]{z}}{\pi^N} \eu^{-\abs{z}^2}
-  \braket{\psi}{z}\!\braket{z}{\psi}
-  \frac{\braket{\psi(t)}{z}\!\mel{z}{L^†\dot{B}(t)}{\psi(t)}}{\braket{\psi}{z}\!\braket{z}{\psi}}
+  \ev{L^†\dot{B}(t)} &= ∫ \frac{\dd[2]{\vb{z}}}{\pi^N} \eu^{-\abs{\vb{z}}^2}
+  \braket{\psi}{\vb{z}}\!\braket{\vb{z}}{\psi}
+  \frac{\braket{\psi(t)}{\vb{z}}\!\mel{\vb{z}}{L^†\dot{B}(t)}{\psi(t)}}{\braket{\psi}{\vb{z}}\!\braket{\vb{z}}{\psi}}
   \\
-  &= ∫ \frac{\dd[2]{z}}{\pi^N} \eu^{-\abs{z}^2}
-  \frac{\mel{z(t)}{L^†\dot{B}(t)}{\psi(t)}}{\braket{\psi}{z(t)}\!\braket{z(t)}{\psi}},
+  &= ∫ \frac{\dd[2]{\vb{z}}}{\pi^N} \eu^{-\abs{\vb{z}}^2}
+  \frac{\mel{\tilde{\vb{z}}(t)}{L^†\dot{B}(t)}{\psi(t)}}{\braket{\psi}{\tilde{\vb{z}}(t)}\!\braket{\tilde{\vb{z}}(t)}{\psi}},
   \end{aligned}
 \end{equation}
-where \(z_{\lambda}^{*}(t)=z_{\lambda}^{*}+\i g_{\lambda} ∫_{0}^{t}
+where \(\tilde{z}_{\lambda}^{*}(t)=z_{\lambda}^{*}+\i g_{\lambda} ∫_{0}^{t}
 \dd{s} \eu^{-\i ω_{\lambda} s}\ev{L^†}_{s}\).
-We find that next steps are the same as in \cref{sec:nonlin} by noting
+
+It has to be shown now, that the term
+\({\braket{\psi}{\tilde{\vb{z}}(t)}\!\braket{\tilde{\vb{z}}(t)}{\psi}}\)
+can be evaluated in the same fashion as in \cref{sec:flow_lin}.  We
+proceed as in \cref{sec:nonlin} by noting
 \begin{equation}
   \label{eq:deriv_trick}
   \begin{aligned}
   \eval{∂_{z^\ast_\lambda}}_{z^\ast=z_\lambda^\ast(t)} &=
-  ∫_0^t\dd{s}\eval{\pdv{η^\ast_s}{z^\ast_\lambda}}_{z^\ast=z^\ast_\lambda(t)}
-                                                         \fdv{}{η^\ast_s(z^\ast=z^\ast(t))} \\
+  ∫_0^t\dd{s}\eval{\pdv{η^\ast_s}{z^\ast_\lambda}}_{\vb{z}^\ast=\vb{z}^\ast(t)}
+                                                         \fdv{}{η^\ast_s(\vb{z}^\ast=\tilde{\vb{z}}^\ast(t))} \\
     &=
-  ∫_0^t\dd{s}\eval{\pdv{η^\ast_s}{z^\ast_\lambda}}_{z^\ast=z^\ast(0)}
-  \fdv{}{η^\ast_s(z^\ast=z^\ast(t))},
+  ∫_0^t\dd{s}\eval{\pdv{η^\ast_s}{z^\ast_\lambda}}_{\vb{z}^\ast=\tilde{\vb{z}}^\ast(0)}
+  \fdv{}{η^\ast_s(\vb{z}^\ast=\tilde{\vb{z}}^\ast(t))},
   \end{aligned}
 \end{equation}
 which does alter the definition of \(D_t\) but results in the same
 HOPS equations.
-The shifted process \(\tilde{η}^\ast=
-η^\ast(z^\ast(t),t)=η^\ast(t) +
+The shifted process \(\tilde{η}^\ast_{t}=
+η_{t}^\ast(\tilde{\vb{z}}^\ast(t),t)=η^\ast_{t} +
 ∫_0^t\dd{s}\alpha^\ast(t-s)\ev{L^†}_{\psi_s}\) appears directly
 in the NMQSD equation but results only in a slight change in the
 functional derivative. Note however that
 \begin{equation}
   \label{eq:fdvclarification}
-  \fdv{}{η^\ast_s(z^\ast=z^\ast(t))} \neq \fdv{}{\tilde{η}^\ast_s}
+  \fdv{}{η^\ast_s(\vb{z}^\ast=\tilde{\vb{z}}^\ast(t))} \neq \fdv{}{\tilde{η}^\ast_s}
 \end{equation}
-which is not problematic as we have (implicit in~\cite{Diosi1998Mar})
-\begin{equation}
-  \label{eq:fdvhops}
-  \fdv{}{η^\ast_s(z^\ast=z^\ast(t))} \ket{\psi(z^\ast)} = \fdv{}{η^\ast_s}\ket{\psi(z^\ast(t, z^\ast_0), t)}
-\end{equation}
-so that the usual HOPS hierarchy follows. Note \(z^\ast_0 = z^\ast(0)\).
+which is not problematic as we absorb the functional derivative into
+the definition of the hierarchy state.
 
 Therefore,
 \begin{equation}
@@ -242,31 +236,20 @@ Therefore,
   J(t) =
   -\i
   \mathcal{M}_{\tilde{η}^\ast}\frac{\mel{\psi(\tilde{η},t)}{L^†\dot{\tilde{D}}_t}{\psi(\tilde{η}^\ast,t)}}{\braket{\psi(\tilde{η},t)}{\psi(\tilde{η}^\ast,t)}}
-  + \cc,
+  + \cc.
 \end{equation}
-where the dependence on \(\tilde{η}\) is symbolic and to be
-understood in the context of \cref{eq:fdvhops}.
-
-Again we express the result in the language of~\cite{Hartmann2021Aug}
-to obtain
-\begin{equation}
-  \label{eq:nonlinhopsflowrich}
-  J(t) = ∑_\mu\frac{G_\mu W_\mu}{\bar{g}_\mu}
-  \i\mathcal{M}_{η^\ast}\frac{\bra{\psi^{(0)}(η,
-      t)}L^†\ket{\psi^{\vb{e}_\mu}(η^\ast,t)}}{\bra{\psi^{(0)}(η,
-      t)}\ket{\psi^{0}(η^\ast,t)}} + \cc.
-\end{equation}
-
-With the new ``fock-space'' normalization however the expression
-becomes
+and in the language of HOPS
 \begin{equation}
   \label{eq:nonlinhopsflowfock}
   J(t) = - ∑_\mu\sqrt{G_\mu}W_\mu
-  \mathcal{M}_{η^\ast}\frac{\bra{\psi^{(0)}(η,
-      t)}L^†\ket{\psi^{\vb{e}_\mu}(η^\ast,t)}}{\bra{\psi^{(0)}(η,
-      t)}\ket{\psi^{0}(η^\ast,t)}} + \cc.
+  \mathcal{M}_{\tilde{η}^\ast}\frac{\bra{\psi^{(0)}(\tilde{η},
+      t)}L^†\ket{\psi^{\vb{e}_\mu}(\tilde{η}^\ast,t)}}{\bra{\psi^{(0)}(\tilde{η},
+      t)}\ket{\psi^{(0)}(\tilde{η}^\ast,t)}} + \cc.
 \end{equation}
 
+In essence, the expressions derived in \cref{sec:flow_lin}  simply
+have to be normalized.
+
 \section{Generalization to Finite Temperature}
 \label{sec:lin_finite}
 \begin{itemize}
@@ -286,7 +269,8 @@ like to recover the usual pure state zero temperature formalism. There
 are multiple methods for dealing with a thermal initial such as the
 thermofield method (see~\cite{Diosi1998Mar}), but because the results
 discussed here are to be applied with the HOPS method we shall use the
-method described in~\cite{Hartmann2017Dec}.
+method described in~\cite{Hartmann2017Dec}, as this is the most mature
+and well tested one.
 
 The shift operator
 \begin{equation}

src/intro.tex

@@ -282,7 +282,9 @@ The statistics of the process follow from interpreting
 system state may then be recovered by averaging over all trajectories
 \begin{equation}
   \label{eq:recover_rho}
-  ρ_{\sys}(t) = \tr_{\bath}\bqty{\ketbra{ψ(t)}} = \mathcal{M}_{η_{t}^\ast}\bqty{ψ_t(η_t)^\dag ψ_t(η^\ast_t)}.
+  ρ_{\sys}(t) = \tr_{\bath}\bqty{\ketbra{ψ(t)}} =
+ ∫{\frac{\dd{\vb{z}}}{π^{N}}\eu^{-\abs{\vb{z}}^2}}\ketbra{ψ(t,\vb{z})}{ψ(t,\vb{z}^\ast)}=
+  \mathcal{M}_{η_{t}^\ast}\bqty{ψ_t(η_t)^\dag ψ_t(η^\ast_t)}.
 \end{equation}
 
 Note that the BCF \(α\) is usually defined as Fourier transform of the
@@ -306,7 +308,16 @@ state, encoded in the value of the stochastic process.
 
 The equation \cref{eq:nmqsd_single} does not preserve the norm of the
 state, leading to suboptimal convergence of \cref{eq:recover_rho}.
-To remedy this, we choose a co-moving shifted stochastic process
+When recovering the system state, we would like to average of
+normalized states
+\begin{equation}
+  \label{eq:norm_av}
+  ρ_{\sys}(t) =
+ ∫{\frac{\dd{\vb{z}}}{π^{N}}\eu^{-\abs{\vb{z}}^2}}
+ \braket{ψ(t,\vb{z})}{ψ(t,\vb{z}^\ast)} \frac{\ketbra{ψ(t,\vb{z})}{ψ(t,\vb{z}^\ast)}}{\braket{ψ(t,\vb{z})}{ψ(t,\vb{z}^\ast)}}.
+\end{equation}
+
+This can be achieved by using a co-moving shifted stochastic process
 \begin{equation}
   \label{eq:shifted_proc}
   \tilde{η}_{t}^\ast= η^\ast_{t} + ∫_{0}^{t}\dd{s} α^\ast(t-s) \ev{L^\ast}_{s},
@@ -316,7 +327,9 @@ where
 ψ(\tilde{η}_{t}^\ast)_{t}\). The origin of this shift lies in the
 study of the Husimi \(Q\) function of the bath
 \(Q_{t}(\vb{z}, \vb{z}^\ast) = \eu^{-\abs{z}^{2}} π^{-N}
-\braket{ψ(t,\vb{z})}{ψ(t,\vb{z}^\ast)}\).
+\braket{ψ(t,\vb{z})}{ψ(t,\vb{z}^\ast)}\). Shifting the process, or on
+a deeper level the stochastic state labels, amounts to importance
+sampling for each time step.
 
 This leads to the nonlinear NMQSD equation
 \begin{equation}
@@ -329,7 +342,7 @@ discussed here, but in \cref{sec:nonlin}.
 Crucially, the system state is now recovered through
 \begin{equation}
   \label{eq:recover_rho_nonlinear}
-  ρ_{\sys}(t) = \mathcal{M}_{η_{t}^\ast}\bqty{\frac{ψ_t(η_t)^\dag ψ_t(η^\ast_t)}{\norm{ψ_t(η_t)}^{2}}},
+  ρ_{\sys}(t) = \mathcal{M}_{\tilde{η}_{t}^\ast}\bqty{\frac{ψ_t(\tilde{η}_t)^\dag ψ_t(\tilde{η}^\ast_t)}{\norm{ψ_t(\tilde{η}_t)}^{2}}},
 \end{equation}
 so that all trajectories contribute with ``equal weight''.
 
@@ -348,8 +361,9 @@ There exist analytical approaches to this
 term~\cite{Diosi1998Mar,Strunz2001Habil}, but we keep the approach as
 general as possible and instead choose a numerical avenue.
 
-They key is define away the complicated term containing the functional
-derivative as an auxiliary state. Expanding the BCF into exponentials
+They key~\cite{Suess2014Oct,Hartmann2017Dec,RichardDiss} is define
+away the complicated term containing the functional derivative as an
+auxiliary state. Expanding the BCF into exponentials
 \(α(τ)=∑_{μ}G_{μ=1}^{M}\eu^{-W_{μ}τ}\) and defining
 \begin{equation}
   \label{eq:d_op_one}