norm estimate updates

tex/hops_tweaks/src/index.tex

@@ -239,7 +239,122 @@ are bosonic Fock-states.
 Now \cref{eq:multihops} becomes
 \begin{equation}
   \label{eq:fockhops}
-  ∂_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 +
-  \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}.
+  \begin{aligned}
+    ∂_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 +
+                 \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}\\
+               &= \tilde{H}\ket{Ψ}
+  \end{aligned}
 \end{equation}
+
+\section{Estimating the Norms of the Auxiliary States}
+\label{sec:normest}
+
+It is possible to find an (semi-rigorous) upper bound to the norms of
+the auxiliary states. We will limit ourselves to one bath. The
+generalization to multiple baths is straight forward.
+
+Using \cref{eq:fockhops}, we can calculate
+\begin{equation}
+  \label{eq:normdiff}
+  \begin{aligned}
+    \iu ∂_t \norm{ψ^{\vb{k}}}^2
+    &= \bra{Ψ}\ket{k}\bra{k}\tilde{H}\ket{Ψ} - \cc\\
+    &= \qty(ψ^{\vb{k}})^†\bra{k}
+      \qty[-\iu L η^\ast -\iu ∑_{μ=1}^{M}b_{μ}^\dag b_{μ} W_μ
+      +∑_{μ=1}^{M} \sqrt{G_{μ}} \qty(b^†_{μ}L +
+      b_μ L^†)]\ket{Ψ}- \cc\\
+    &= \Bigg[-\iu \qty(ψ^{\vb{k}})^†L η^\ast ψ^{\vb{k}}
+        -\iu ∑_{μ=1}^{M}k_μ W_μ \norm{ψ^{\vb{k}}}^2\\
+        &\phantom{=}\quad -∑_{μ=1}^{M}\qty[\qty(ψ^{\vb{k}})^†\sqrt{G_{μ}k_μ}Lψ^{\vb{k}-\vb{e}_μ} +
+        \qty(ψ^{\vb{k}})^†\sqrt{G_{μ}(k_μ+1)}Lψ^{\vb{k}+\vb{e}_μ} ]\Bigg]  - \cc.
+  \end{aligned}
+\end{equation}
+
+We can now further treat the this expression to find the steady state
+norms of the states.
+
+Assuming generically that the term containing the stochastic process
+\(η\) vanishes in the time average (as is the case for the steady
+state) we will drop it in the following.
+
+Terms of the form \(\Im(ψ^† O φ)\) may be estimated as follows
+\begin{equation}
+  \label{eq:genericest}
+  \abs{\Im(ψ^† O φ)} \leq \norm{ψ} \norm{O φ} \leq \norm{ψ}\norm{O}\norm{φ},
+\end{equation}
+where the norm on the operator is the standard linear operator norm
+\(\norm{O} = \max_{x\in \mathcal{H}}\frac{\ev{O}{x}}{\braket{x}}\).
+
+We now endeavor to find from \cref{eq:normdiff} an estimate of the
+steady state norm of \(ψ^{\vb{k}}\). To this end we assume that the
+coupling to higher hierarchy states generically lowers the norm and is
+therefore neglected. Using \cref{eq:genericest} we can estimate the
+influence of the coupling to lower states, choosing the sign
+so that the contribution to the norm is positive.
+
+With this we obtain
+\begin{equation}
+  \label{eq:finalest}
+  ∂_t \norm{ψ^{\vb{k}}}^2 = 0 = -∑_{μ=1}^{M}k_μ \Re[W_μ]
+  \norm{ψ^{\vb{k}}}^2 +
+  ∑_{μ=1}^{M}\abs{\sqrt{G_{μ}k_μ}}\norm{ψ^{\vb{k}}}\norm{ψ^{\vb{k}-\vb{e}_μ}}\norm{L}
+\end{equation}
+and therefore
+\begin{equation}
+  \label{eq:steadynorm}
+  \norm{ψ^{\vb{k}}} =
+  \frac{∑_{μ=1}^{M}\abs{\sqrt{G_{μ}k_μ}}\norm{ψ^{\vb{k}-\vb{e}_μ}}\norm{L}}{∑_{μ=1}^{M}k_μ \Re[W_μ]}.
+\end{equation}
+
+
+For the nonlinear method, the stochastic process obtains a shift whose
+magnitude can be estimated as follows
+\begin{equation}
+  \label{eq:shiftestimate}
+  \abs{η_{\mathrm{sh}}} \leq \norm{L} ∫_0^∞\dd{s}\abs{α^\ast(t-s)} \leq
+  \norm{L} \sum_{μ=1}^M \frac{\abs{G_μ}}{\Re[W_μ]}.
+\end{equation}
+
+Assuming the contribution of the shift is norm enhancing, we arrive at
+the expression
+\begin{equation}
+  \label{eq:steadynorm_nonlin}
+  \norm{ψ^{\vb{k}}} =
+  \frac{∑_{μ=1}^{M}\abs{\sqrt{G_{μ}k_μ}}\norm{ψ^{\vb{k}-\vb{e}_μ}}\norm{L}}{∑_{μ=1}^{M}k_μ
+    \Re[W_μ] - \norm{L}^2 \sum_{μ=1}^M \frac{\abs{G_μ}}{\Re[W_μ]}}.
+\end{equation}
+
+This indicates, that we trade hierarchy depth for sample count in the
+nonlinear method. The Markov limit may be related to
+\(\Re[W_μ]\rightarrow ∞\). In this case the extra term vanishes and
+the nonlinear method looses its advantage (as the shift vanishes).
+
+The relations \cref{eq:steadynorm,eq:steadynorm_nonlin} are recursive
+and break off at \(ψ^0\), the norm of which can be assumed to be \(1\)
+in the nonlinear method.
+
+
+These ideas remain to be verified. Especially the assumptions should
+be checked. For time dependent coupling, one may maximize the estimate
+over all \(L(t)\).
+
+\subsection{Truncation Scheme}
+\label{sec:truncsch}
+The norm of the \(\vb{k}\)th hierarchy state scales like
+\({1} / {\sqrt{\max_μk_μ}}\). This fact in itself, however, is not
+too meaningful as the magnitude of the coupling to the lower hierarchy
+states is
+\begin{equation}
+  \label{eq:couplingmag}
+  M_{\vb{k}} = \norm{L} \norm{ψ^{\vb{k}}} \max_μ \abs{\sqrt{G_μk_μ}},
+\end{equation}
+which balances out the scaling.
+
+Calculating \(M_{\vb{k}}\) explicitly and demanding it to be small
+(compared to some energy scale) nevertheless gives a convergent
+truncation scheme below a certain coupling strength.
+
+Some basic experimentation has shown, that the cutoff parameter has to
+be tuned and is not universally valid.