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src/analytical_solution.tex
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\chapter{Comparison with an Analytical Solution}
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\section{One Oscillator, One Bath}
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\label{sec:oneosc}
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\subsection{Model}
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\label{sec:model}
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The model is given by the quadratic hamiltonian
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\begin{equation}
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\label{eq:hamiltonian}
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H = \frac{Ω}{4}\qty(p^2+q^2) + \frac{1}{2} q
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\sum_λ\qty(g_λ^\ast b_λ + g_λ
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b^†_λ)+\sum_λ\omega_λ b^†_λ b_λ,
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\end{equation}
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where \(a,a^†\) are the ladder operators of the harmonic
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oscillator, \(q=a+a^†\) and \(p=\frac{1}{\iu}\qty(a-a^†)\) so
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that \([q,p] = 2\iu\).
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\subsection{Equations of Motion}
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\label{sec:eqmot}
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The Heisenberg equation yields
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\begin{align}
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\dot{q} &=Ω p \label{eq:qdot}\\
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\dot{p} &= -Ω q - \int_0^t \Im[α_0(t-s)] q(s)\dd{s} + W(t) \label{eq:pdot}
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\\
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\dot{b}_λ &= -\iu g_λ \frac{q}{2} - \iu\omega_λ b_λ
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\end{align}
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with the operator noise
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\(W(t)=-\sum_λ \qty(g_λ^\ast b_λ(0)
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\eu^{-\iu\omega_λ t } + g_λ b_λ^†(0)
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\eu^{\iu\omega_λ t })\),
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\(\ev{W(t)W(s)}=α(t-s)\) and \(α_0 \equiv \eval{α}_{T=0}\).
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The equation \cref{eq:pdot} arises from
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\begin{equation}
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\label{eq:bsol}
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b_λ(t) = b_λ(0) \eu^{-\iu ω_λ t} - \frac{\iu g_λ}{2}∫_0^t
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q(s) \eu^{-\iu ω_λ (t-s)}\dd{s}.
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\end{equation}
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The equations for \cref{eq:qdot} and \cref{eq:pdot} can be solved by
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finding a matrix \(G(t)\) with \(G(0)=\id\) and
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\begin{equation}
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\label{eq:eqmotprop}
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\dot{G}(t) = A G(t) - \int_0^t K(t-s) G(s)\dd{s},\quad A=\mqty(0 &
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Ω \\ -Ω & 0), \quad K(t)=\mqty(0 & 0\\ \Im[α_0(t)] & 0).
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\end{equation}
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Then
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\begin{equation}
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\label{eq:qpsol}
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\mqty(q(t)\\ p(t)) = G(t)\mqty(q(0)\\ p(0)) + \int_0^tG(t-s)
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\mqty(0\\ W(s))\dd{s}.
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\end{equation}
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Because we are only interested in solutions for \(t\geq 0\) and the
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shape of the convolution in \cref{eq:eqmotprop} the solution may be
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found by virtue of the Laplace transform.
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Setting
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\begin{equation}
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\label{eq:laplprop}
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\mathcal{L}\{G\}(z) = \int_0^\infty \eu^{-z\cdot t} G(t)
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\end{equation}
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leads to an algebraic formula
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\begin{equation}
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\label{eq:galgebr}
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\mathcal{L}\{G\}(z) = \qty(z-A + \mathcal{L}\{K\}(z))^{-1}.
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\end{equation}
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\subsection{Solution}
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\label{sec:solution}
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We observe that
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\begin{equation}
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\label{eq:mdef}
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M = z-A + \mathcal{L}\{K\}(z) = \mqty(z & -Ω\\ Ω +
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\mathcal{L}\{\Im[α_0]\}(z) & z)
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\end{equation}
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and therefore
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\begin{equation}
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\label{eq:minv}
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M^{-1} = \frac{1}{Ω^2 + Ω\mathcal{L}\{\Im[α_0]\}(z) + z^2}
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\mqty(z & Ω \\ -(Ω + \mathcal{L}\{\Im[α_0]\}(z)) & z).
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\end{equation}
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From this we can conclude that \(G_{11}=G_{22}\).
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Because \(\ev{W(s)}=0\) for thermal initial states of the bath we have
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\begin{equation}
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\label{eq:meanvals}
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\mqty(\ev{q(t)}\\ \ev{p(t)}) = G(t)\mqty(\ev{q(0)}\\ \ev{p(0)}).
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\end{equation}
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Knowing this, we can deduce from \(\ev{\dot{q}}= Ω \ev{p}\) that
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\begin{align}
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\label{eq:onlyoneneeded}
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G_{11} &= \frac{\dot{G}_{12}}{Ω} & G_{21} &=\frac{\ddot{G}_{12}}{Ω^2}.
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\end{align}
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These relations are true independent of the initial state of the
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system. It therefore suffices if we concern ourselves with
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\(G_{12}\). We nevertheless continue in full generality.
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Assume that \(α_0(t)=\sum_{n=1}^N G_n \eu^{-W_n t - \i
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\varphi_n}\) with \(W_n=\gamma_n + \i\delta_n\) and \(G_n,
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\varphi_n, \gamma_n,\delta_n\in\RR\) for \(t\geq 0\).
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This leads to a mathematically simple expression for the Laplace
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transform
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\begin{equation}
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\label{eq:laplace_alpha}
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\mathcal{L}\qty{\Im[α_0]}(z) = -\sum_n G_n\qty[\frac{(z+\gamma_n)\sin\varphi_n+\delta_n\cos\varphi_n}{\delta_n^2+(\gamma_n+z)^2}].
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\end{equation}
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Because \(\mathcal{L}\{\Im[α_0]\}\) appears in the denominator of
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\cref{eq:minv} it is desirable to write \cref{eq:laplace_alpha} with a
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common denominator. Introducing \(s_n = \sin\varphi_n,\, c_n =
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\cos\varphi_n\) and \(z_n= -W_k\) we arrive
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at
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\begin{equation}
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\label{eq:laplace_alpha_better}
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\begin{aligned}
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\mathcal{L}\qty{\Im[α_0]}(z) &= - \sum_n
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G_n\frac{(z_n+\gamma_n)s_n+ \delta_nc_n}{(z-z_n)(z-z_n^\ast)} \\
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&= -\frac{\sum_n G_n \qty((z_n+\gamma_n)s_n+
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\delta_nc_n)\prod_{k\neq
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n}(z-z_k)(z-z_k^\ast)}{\prod_{k}(z-z_k)(z-z_k^\ast)} \\
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&= \frac{\sum_n f_n(z)\prod_{k\neq n}(z-z_k)(z-z_k^\ast)}{\prod_{k}(z-z_k)(z-z_k^\ast)}
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\end{aligned}
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\end{equation}
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with the polynomials of first degree
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\(f_n(z)=-G_n \qty((z_n+\gamma_n)s_n+\delta_nc_n)\). Because the
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above expression is a rational function, the components of
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\cref{eq:minv} are rational functions for which the Laplace transform
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is particularly simple to invert using the residue theorem. With this
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in mind we now calculate
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\begin{equation}
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\label{eq:prefactorrational}
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\frac{1}{Ω^2 + Ω\mathcal{L}\{\Im[α_0]\}(z) + z^2}
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% =\frac{\prod_{k}(z-z_k)(z-z_k^\ast)}{\qty[(z+\iΩ)(z-\iΩ)]\prod_{k}(z-z_k)(z-z_k^\ast)
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% + \sum_nΩ f_n(z)\prod_{k\neq n}(z-z_k)(z-z_k^\ast)}\\
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=\frac{f_0(z)}{p_1(z) + \sum_n q_n(z)}
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=
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\frac{f_0(z)}{\mu\prod_{n=1}^{N+1}(z-\tilde{z}_l)(z-\tilde{z}^\ast_l)}
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= \frac{f_0(z)}{p(z)}
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\end{equation}
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where
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\begin{align}
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f_0(z) &= \prod_{k}(z-z_k)(z-z_k^\ast) \\
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p_1(z) &= \qty[(z+\iΩ)(z-\iΩ)]\prod_{k}(z-z_k)(z-z_k^\ast) \\
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q_n(z) &= Ω f_n(z)\prod_{k\neq n}(z-z_k)(z-z_k^\ast)
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\end{align}
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and \(\mu\in\RR\). The \(\tilde{z}_l\) are the roots of the real
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polynomial
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\begin{equation}
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\label{eq:polyp}
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p(z) = p_1(z) + \sum_{n=1}^{N}q_n(z)
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\end{equation}
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of degree \(2(N+1)\) where we \textbf{assume that there are
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no roots with multiplicity greater than one}.
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With this we can now calculate the inverse laplace transform of
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expressions of the form \(\frac{f_0(z)g(z)}{p(z)}\) where \(g(z)\) is
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any holonome function so that \(\frac{f_0(z)g(z)}{p(z)} \eu^{z\cdot
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t}\) falls off fast enough for \(t\geq 0\),
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\(\Re(z)>\max_l{\Re(\tilde{z}_l)}=\Delta\) and \(\Re(z) \rightarrow
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-\infty\). With this we can close the contour of the inverse Laplace
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transform
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\begin{equation}
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\label{eq:invlap}
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\mathcal{L}^{-1}\qty{\frac{f_0(z)g(z)}{p(z)}}(t) =
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\frac{1}{2\pi\i}\int_{\Delta - \i\infty}^{\Delta + \i\infty} \frac{f_0(z)g(z)}{p(z)} \eu^{z\cdot
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t}\dd{z}
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\end{equation}
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to the left to obtain
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\begin{equation}
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\label{eq:simpleinvtrans}
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\mathcal{L}^{-1}\qty{\frac{f_0(z)g(z)}{p(z)}}(t)
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=
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\sum_{l=1}^{N+1}\qty[\frac{f_0(\tilde{z}_l)g(\tilde{z}_l)}{p'(\tilde{z}_l)}
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\eu^{\tilde{z}_l \cdot t} + \cc]
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\end{equation}
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where we assumed that \(g(z)^\ast=g(z^\ast)\) which is the case for
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all our purposes. For completeness we give
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\begin{equation}
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\label{eq:pderiv}
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p'(z) = 2\mu\sum_{k=1}^{N+1}\qty[(z-\Re(\tilde{z}_k))\prod_{\substack{n=1\\
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n\neq k}}^{N+1}(z-\tilde{z}_n)(z-\tilde{z}^\ast_n)].
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\end{equation}
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We can immediately conclude that all elements of \(G\) are sums of
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exponentials, just like the BCF. In particular
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\begin{equation}
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\label{eq:gfinal}
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G(t) = \sum_{l=1}^{N+1}\qty[R_l \mqty(\tilde{z}_l & Ω \\ \frac{\tilde{z}_l^2}{Ω} & \tilde{z}_l)\eu^{\tilde{z}_l \cdot
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t} + \cc]
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\end{equation}
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with \(R_l={f_0(\tilde{z}_l)}/{p'(\tilde{z}_l)}\).
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\subsubsection{Negative Times}
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The solution detailed above is only valid for positive times. Because
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we strive to employ the same formalism again for negative times, we
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will concern ourselves with the transformed quantities
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\(\bar{X}(τ) = X(t(τ)) = X(-τ)\) so that \(τ ≥ 0\). It follows that
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\(∂_τ \bar{X}(τ) = \bar{X}'(τ) = -\dot{X}(t(τ))\) so that
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\begin{align}
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\bar{q}' &= -Ω \bar{p} \label{eq:qtag}\\
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\bar{p}' &= Ω \bar{q} + ∫_0^τ \Im[α_0(τ-s)] \bar{q}(s)\dd{s} - \bar{W}(τ) \label{eq:ptag}
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\\
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\bar{b}'_λ &= \iu g_λ \frac{q'}{2} + \iu\omega_λ b'_λ.
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\end{align}
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This leads to an equation for \(\bar{G}(τ)\), namely
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\begin{equation}
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\label{eq:eqmotpropbar}
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\bar{G}'(τ) = -A \bar{G}(τ) + \int_0^τ K(τ-s) \bar{G}(s)\dd{s}.
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\end{equation}
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The solution is obtained from the \(t\geq 0\) case by substituting
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\(A\rightarrow -A\) and \(K\rightarrow -K\).
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We obtain for \(t\leq 0\)
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\begin{equation}
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\label{eq:gfinalbar}
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\bar{G}(τ) = G(-τ) = G(t) = \sum_{l=1}^{N+1}\qty[R_l \mqty(\tilde{z}_l & -Ω \\ -\frac{\tilde{z}_l^2}{Ω} & \tilde{z}_l)\eu^{-\tilde{z}_l \cdot
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t} + \cc]
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\end{equation}
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and
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\begin{equation}
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\label{eq:qpsolneg}
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\mqty(q(t)\\ p(t)) = G(t)\mqty(q(0)\\ p(0)) - \int_t^0 G(t-s)
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\mqty(0\\ W(s))\dd{s}.
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\end{equation}
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\subsection{Applications}
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\label{sec:applications}
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Knowing \(G\) and \(α\), we can calculate all observables of the
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system. Simple closed form expressions of sums of exponentials can be
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obtained by using an exponential expansions of \(α\). Throughout,
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we assume a thermal bath initial state so that \(\ev{W(t)}=0\).
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\subsubsection{Correlation Functions}
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\label{sec:correl}
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We proceed to calculate \(\ev{q(t)q(s)}\). For brevity we set
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\(A=G_{11}\), \(B=G_{12}\), \(p_0=p(0)\) and \(q_0=q(0)\). Then,
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\begin{equation}
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\label{eq:qcorrel}
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\ev{q(t)q(s)} =
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\begin{aligned}[t]
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& A(t)A(s) \ev{q_0^2} + B(t)B(s) \ev{p_0^2} +
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A(t)B(s)\ev{q_0p_0} + B(t)A(s)\ev{p_0q_0} \\
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& +\underbrace{∫_0^t\dd{l}∫_0^s\dd{r} B(t-l)B(s-r)α(l-r)}_{\equiv Λ(t, s)}.
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\end{aligned}
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\end{equation}
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For a pure harmonic oscillator initial state \(\ket{n}\) we have
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\begin{equation}
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\label{eq:hoexp}
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\begin{aligned}
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\ev{q^2_0} &= \ev{p^2_0} = 2n+1 & \ev{q_0p_0} &= \iu.
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\end{aligned}
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\end{equation}
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Note that \(p\) and \(p\) differ from the usual definition by a factor
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of \(2\).
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\subsubsection{Bath Enery Derivative}
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\label{sec:bathflow}
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With \cref{eq:qcorrel} we can calculate the time derivative of the
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bath energy expectation value
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\begin{equation}
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\label{eq:bathderiv_1}
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\begin{aligned}
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\ev{\dot{H}_B} &= ∑_λ ω_λ \qty(\ev{b_λ^†\dot{b}_λ} + \cc) \\
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&=\frac{1}{4}∫_0^t\dd{s}\qty[\ev{q(s)q(t)} ∑_λ\abs{g_λ}^2 \eu^{\i
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ω_λ(t-s)} + \cc] +
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\i ∫_0^t\dd{s} G_{12}(s)∑_λ\abs{g_λ}\bar{n}_λ\qty[\eu^{\i ω_λ s}+\cc]\\
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&= -\frac{1}{2}\Im\qty[∫_0^t\dd{s}\ev{q(t)q(s)}\dot{α}_0(t-s)] +
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\frac{1}{2}∫_0^t\dd{s} G_{12}(s)\partial_s\qty[α(s)-α_0(s)] \\
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&=
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\begin{aligned}[t]
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-\frac{1}{2}\Im&\qty[∫_0^t\dd{s}\ev{q(t)q(s)}\dot{α}_0(t-s)] \\&+
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\frac12 G_{12}(t)[α(t)-α_0(t)]
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-\frac{Ω}{2}∫_0^t\dd{s} G_{11}(s)\qty[α(s)-α_0(s)],
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\end{aligned}
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\end{aligned}
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\end{equation}
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where we've used \(\ev{b_λ(0)}=\ev{b_λ^0}=0\),
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\begin{equation}
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\label{eq:blambdadotexp}
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\begin{aligned}
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\ev{b_λ^†\dot{b}_λ}= -\i\ev{b_λ^†\qty(\frac{g_λ}{2}q + ω_λb_λ)} =
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-\i\qty[\frac{g_λ}{2}\ev{b^†_λ(t)q(t)} +
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\underbrace{ω_λ\ev{b^†_λ(t)b_λ(t)}}_{\in\RR\implies\text{cancels }\cc}].
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\end{aligned}
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\end{equation}
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and
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\begin{equation}
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\label{eq:moreladot}
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\begin{aligned}
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\ev{b^†_λ(t)q(t)} &= \ev{\qty(b(0)^{\dag}_λ\eu^{\i ω_λ t} +
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\frac{\i}{2}∫_0^tg_λ^\ast q(s)\eu^{\i ω_λ (t-s)}\dd{s})q(t)} \\
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&= \frac{\i g_λ^\ast}{2}∫_0^t\ev{q(s)q(t)\eu^{\i ω_λ(t-s)}}\dd{s}
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- g_λ^\ast\bar{n}_λ∫_0^t G_{12}(s)\eu^{\i ω_λ s}\dd{s}
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\end{aligned}
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\end{equation}
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with \(\bar{n}_λ=\ev{b^†_λ(0)b_λ(0)}\).
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For further evaluation of \cref{eq:bathderiv_1} we have to calculate
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\begin{equation}
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\label{eq:lambdafold}
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\begin{aligned}
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Λ(t)&=∫_0^t\dd{s}Λ(t,s)\dot{α}_0(t-s)
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=∫_0^t\dd{s}∫_0^t\dd{l}∫_0^s\dd{r}
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B(t-l)B(s-r)α(l-r)\dot{α}_0(t-s)\\
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&=∫_0^t\dd{s}∫_0^t\dd{r}
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\begin{aligned}[t]
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Θ(s-r)&B(s-r)\dot{α}_0(t-s)\times\\
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\biggl[
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&∫_0^{t-r}\dd{u}B(t-r-u)α(u)+∫_0^{r}\dd{u}B(t-r+u)α^\ast(u)
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\biggr]
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\end{aligned}
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\\
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&= ∫_0^t\dd{r} g_1(t-r)\qty[g_2(t-r) + g_3(t,r)] = Λ_1(t) + Λ_2(t)
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\end{aligned}
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\end{equation}
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This expression now only uses \(α(t)\) for \(t\geq 0\) so that we can
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once again employ the exponential expansion for \(α\). In fact, all
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other quantities in \cref{eq:lambdafold} have exponential expansion so
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that we can now define\footnote{Note that this is inconsistent with
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||||
\cref{sec:solution}.}
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\begin{equation}
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||||
\label{eq:expansions}
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||||
\begin{aligned}
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||||
α_0&=∑_k U_k\eu^{-Q_k t} & \dot{α}_0&=∑_k P_k\eu^{-L_k t} & α(t)
|
||||
&= ∑_nG_n\eu^{-W_n t} \\
|
||||
A(t) &= ∑_l A_l\eu^{-C_l t} & B(t) &= ∑_l B_l\eu^{-C_l t}.
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
With this we can calculate,
|
||||
\begin{align}
|
||||
\label{eq:lambdaintegrals}
|
||||
∫_r^t\dd{s}B(s-r)\dot{α}_0(t-s)
|
||||
&=\sum_{m,k}\underbrace{\frac{B_mP_k}{L_k-C_m}}_{\equiv
|
||||
Γ^1_{mk}}\qty[\eu^{-C_m(t-r)}-\eu^{-L_k(t-r)}]=g_1(t-r)\\
|
||||
∫_0^{t-r}\dd{u}B(t-r-u)α(u)
|
||||
&=\sum_{n,l}\underbrace{\frac{B_nG_l}{C_n-W_l}}_{\equiv
|
||||
Γ^2_{nl}}\qty[\eu^{-W_l(t-r)}-\eu^{-C_n(t-r)}]=g_2(t-r)\\
|
||||
∫_0^{r}\dd{u}B(t-r+u)α^\ast(u)
|
||||
&=\sum_{n,l}\underbrace{\frac{B_nG_l^\ast}{C_n+W_l^\ast}}_{\equiv
|
||||
Γ^3_{nl}}\qty[\eu^{-C_n(t-r)}-\eu^{-W_l^\ast r-C_n t}]=g_3(t,r)
|
||||
\end{align}
|
||||
and
|
||||
\begin{align}
|
||||
\label{eq:finalsummands}
|
||||
Λ_1(t)&= ∑_{m,k,n,l}Γ^1_{mk}Γ^2_{nl}\qty[\frac{1-\eu^{-(C_m+W_l)t}}{C_m+W_l}
|
||||
-
|
||||
\frac{1-\eu^{-(C_m+C_n)t}}{C_m+C_n}-
|
||||
\frac{1-\eu^{-(L_k+W_l)t}}{L_k+W_l}
|
||||
+
|
||||
\frac{1-\eu^{-(L_k+C_n)t}}{L_k+C_n}]\\
|
||||
Λ_2(t)&=
|
||||
\begin{aligned}[t]
|
||||
∑_{m,k,n,l}Γ^1_{mk}Γ^3_{nl}\biggl[\frac{1-\eu^{-(C_m+C_n)t}}{C_m+C_n}
|
||||
&-\frac{1-\eu^{-(L_k+C_n)t}}{L_k+C_n}
|
||||
\\&-\frac{\eu^{-(C_n+W_l^\ast)t}-\eu^{-(C_m+C_n)t}}{C_m-W_l^\ast}
|
||||
+\frac{\eu^{-(C_n+W_l^\ast)t}-\eu^{-(L_k+C_n)t}}{L_k-W^\ast_l}\biggr]
|
||||
\end{aligned}
|
||||
\end{align}
|
||||
|
||||
Also required for \cref{eq:bathderiv_1} are
|
||||
\begin{align}
|
||||
\label{eq:ABconv}
|
||||
∫_0^t\dd{s}A(s)\dot{α}_0(t-s) &= ∑_{n,m}\underbrace{\frac{A_nP_m}{L_m-C_n}}_{\equiv
|
||||
Γ^A_{nm}}\qty[\eu^{-C_n t}-\eu^{-L_m t}]\\
|
||||
∫_0^t\dd{s}B(s)\dot{α}_0(t-s) &= ∑_{n,m}Γ^1_{nm}\qty[\eu^{-C_n t}-\eu^{-L_m t}]
|
||||
\end{align}
|
||||
and
|
||||
\begin{multline}
|
||||
\label{eq:nonzerotemplim}
|
||||
∫_0^t\dd{s}A(s)\qty(α(s)-α_0(s)) =\\
|
||||
∑_{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 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}%
|
||||
\label{sec:twoosc}
|
||||
|
||||
The considerations of~\cref{sec:oneosc} can be straight forwardly
|
||||
generalized to the case of two coupled oscillators coupled in turn to
|
||||
a bath each.
|
||||
|
||||
We will not give explicit formulas for the results in terms of sums of
|
||||
exponentials, as they are quite extensive and easily obtained via the
|
||||
use of a computer algebra system or the aforementioned code.
|
||||
|
||||
\subsection{Model}
|
||||
\label{sec:twomodel}
|
||||
|
||||
The model is again given by a quadratic hamiltonian
|
||||
\begin{equation}
|
||||
\label{eq:hamiltonian}
|
||||
\begin{aligned}
|
||||
H &= ∑_{i\in\qty{1,2}} \qty[H^{(i)}_O + q_iB^{(i)} + H_B^{(i)}] + \frac{γ}{4}(q_1-q_2)^2,
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
where \(H_O^{(i)}= \frac{Ω_i}{4}\qty(p_i^2+q_i^2)\), \(B^{(i)}=\sum_λ\qty(g^{(i),\ast}_λb^{(i)}_λ + g^{(i)}_λ
|
||||
b^{(i),†}_λ)\) and \(H_B^{(i)}=\sum_λ\omega_λ b^{(i),†}_λ b^{(i)}_λ\).
|
||||
|
||||
The \(b^{(i)}\) are the usual bosonic ladder operators of the baths.
|
||||
The \(a_i^{(i)},a_i^{†}\) are the ladder operators of the harmonic
|
||||
oscillators and \(q_i=a_i+a_i^†\) and \(p=\frac{1}{\iu}\qty(a_i-a_i^†)\) so
|
||||
that \([q_i,p_j] = 2\iuδ_{ij}\) and \([q_i,q_j] = [p_i,p_j] = 0\).
|
||||
|
||||
\subsection{Equations of Motion}
|
||||
\label{sec:eqmot_two}
|
||||
The Heisenberg equation yields
|
||||
\begin{align}
|
||||
\dot{q}_i &= Ω_i p_i \label{eq:qidot}\\
|
||||
\dot{p}_i &= -(Ω_i+γ) q_i - \int_0^t \Im[α^{(i)}_0(t-s)] q_i(s)\dd{s} + W_i(t) \label{eq:pidot}
|
||||
\\
|
||||
\dot{b}^{(i)}_λ &= -\iu g^{(i)}_λ \frac{q_i}{2} - \iu\omega^{(i)}_λ b^{(i)}_λ
|
||||
\end{align}
|
||||
|
||||
with the operator noise
|
||||
\(W_i(t)=-\sum_λ \qty(g_λ^{(i),\ast} b^{(i)}_λ(0)
|
||||
\eu^{-\iu\omega^{(i)}_λ t } + g_λ^{(i)} b_λ^{(i),†}(0)
|
||||
\eu^{\iu\omega^{(i)}_λ t })\) satisfying \(\ev{W_i(s)}=0\) and
|
||||
\(\ev{W_i(t)W_j(s)}=δ_{ij}α^{(i)}(t-s)\). We introduced \(α^{(i)}_0
|
||||
\equiv \eval{α^{(i)}}_{T=0}\).
|
||||
|
||||
We have given most quantities an extra index and accounted for the
|
||||
coupling between the two oscillators. Apart from this, the equations
|
||||
of motion have the same structure as in \cref{seq:eqmot}.
|
||||
|
||||
Again, we obtain
|
||||
\begin{equation}
|
||||
\label{eq:bsoltwo}
|
||||
b^{(i)}_λ(t) = b^{(i)}_λ(0) \eu^{-\iu ω^{(i)}_λ t} - \frac{\iu g^{(i)}_λ}{2}∫_0^t
|
||||
q_i(s) \eu^{-\iu ω^{(i)}_λ (t-s)}\dd{s}.
|
||||
\end{equation}
|
||||
|
||||
We can solve the equations for the \(q_i,\,p_i\)
|
||||
by finding a matrix \(G(t)\) with \(G(0)=\id\) and
|
||||
\begin{gather}
|
||||
\label{eq:eqmotproptwo}
|
||||
\dot{G}(t) = A G(t) - \int_0^t K(t-s) G(s)\dd{s}\\
|
||||
A = \mqty(
|
||||
0 & \Omega & 0 & 0 \\
|
||||
-\gamma -\Omega & 0 & \gamma & 0 \\
|
||||
0 & 0 & 0 & \Lambda \\
|
||||
\gamma & 0 & -\gamma -\Lambda & 0),\;
|
||||
K(τ) =
|
||||
\mqty(0 & 0 & 0 & 0 \\
|
||||
\Im[α^{(1)}_0(t-s)] & 0 & 0 & 0 \\
|
||||
0 & 0 & 0 & 0 \\
|
||||
0 & 0 & \Im[α^{(2)}_0(t-s)] & 0),
|
||||
\end{gather}
|
||||
where \(Ω=Ω_1\) and \(Λ=Ω_2\) for convenience.
|
||||
|
||||
Then
|
||||
\begin{equation}
|
||||
\label{eq:qpsol}
|
||||
\mqty(q_1(t)\\ p_1(t)\\ q_2(t)\\ p_2(t)) = G(t)\mqty(q_1(0)\\ p_1(0) \\ q_2(0)\\ p_2(0)) + ∫_0^tG(t-s)
|
||||
\mqty(0\\ W_1(s)\\ 0 \\ W_2(s))\dd{s}.
|
||||
\end{equation}
|
||||
|
||||
With the Laplace transform find for \(t\geq 0\) the formula
|
||||
\cref{eq:galgebr} for the Laplace transform of the solution, albeit
|
||||
now with a more complicated matrix
|
||||
\begin{equation}
|
||||
\label{eq:mdeftwo}
|
||||
M = z-A + \mathcal{L}\{K\}(z) = \mqty(z & -\Omega & 0 & 0 \\
|
||||
\mathcal{L}\{\Im[α^{(1)}_0]\}(z)+\gamma +\Omega & z & -\gamma & 0 \\
|
||||
0 & 0 & z & -\Lambda \\
|
||||
-\gamma & 0 & \mathcal{L}\{\Im[α^{(2)}_0]\}(z)+\gamma +\Lambda & z)
|
||||
\end{equation}
|
||||
that we have to invert.
|
||||
|
||||
This can be done easily\footnote{We have use a computer algebra
|
||||
system. There is probably a pattern to the inverse matrix which can
|
||||
be found so that the solution for \(N>2\) oscillators may be found.}
|
||||
and yields
|
||||
\begin{equation}
|
||||
\label{eq:minvtwo}
|
||||
M^{-1}(z) = \frac{1}{\det[M](z)} \tilde{M}(z)
|
||||
\end{equation}
|
||||
where \(\tilde{M}\) is a matrix containing only polynomials of \(z\)
|
||||
and of the Laplace transforms of the bath correlation functions.
|
||||
|
||||
The numerator
|
||||
\begin{equation}
|
||||
\label{eq:numerator}
|
||||
\begin{aligned}
|
||||
\det[M](z)=a& b \Lambda \Omega +a \left(\gamma
|
||||
\Lambda \Omega +\Lambda ^2 \Omega +\Omega z^2\right)
|
||||
+b
|
||||
\left(\gamma \Lambda \Omega +\Lambda \Omega ^2+\Lambda z^2\right)\\
|
||||
&+\gamma \Lambda ^2 \Omega +\gamma \Lambda
|
||||
\Omega ^2+\Lambda ^2 \Omega ^2+\gamma \Lambda z^2+\gamma \Omega z^2+\Lambda ^2 z^2+\Omega ^2 z^2+z^4
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
where \(a=\mathcal{L}\{\Im[α^{(1)}_0]\}\) and \(b=\mathcal{L}\{\Im[α^{(2)}_0]\}\).
|
||||
|
||||
Using the same approach as in \cref{sec:solution}, we arrive at an
|
||||
expression similar to \cref{eq:prefactorrational} for
|
||||
\((\det[M](z))^{-1}\). The polynomial \(p\) is now of degree
|
||||
\(4 + 2 \qty(N^{(1)} + N^{(j)})\) where the \(N^{(i)}\) are the number of
|
||||
terms in the expansions of bath correlation functions for each bath
|
||||
and the function \(f_0\) now depends on both bath correlation
|
||||
functions.
|
||||
|
||||
We ultimately find that \(G\) is a sum of
|
||||
exponentials
|
||||
\begin{equation}
|
||||
\label{eq:gfinal}
|
||||
G(t) = \sum_{l=1}^{2+N_1+N_2}\qty[R_l \tilde{M}(\tilde{z}_l)\eu^{\tilde{z}_l \cdot
|
||||
t} + \cc]
|
||||
\end{equation}
|
||||
with \(R_l={f_0(\tilde{z}_l)}/{p'(\tilde{z}_l)}\).
|
||||
|
||||
\subsection{Applications}
|
||||
\subsubsection{Correlation Functions}
|
||||
\label{sec:correltwo}
|
||||
|
||||
We can now proceed to calculate the correlation functions
|
||||
\(C(t,s) = \ev{x_i(t)x_j(s)}\) where the \(x_i\) are the phase space operators
|
||||
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)_{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}
|
||||
|
||||
The matrix \(Θ_{ij}\) contains the bath-induced correlations and can
|
||||
be calculated as in the single-oscillator case.
|
||||
|
||||
|
||||
\subsubsection{Bath Enery Derivative}
|
||||
\label{sec:bathflowtwo}
|
||||
|
||||
Similar to the calculations in \cref{sec:bathflow} we find
|
||||
\begin{equation}
|
||||
\label{eq:bathderivtwo}
|
||||
\ev{\dot{H}_B^{(n)}}=-\frac12
|
||||
\Im∫_0^tC_{2n-1, 2n-1}(t,s)\dot{α}_0^{(n)}(t-s)\dd{s} + \frac12 ∫_0^t
|
||||
∑_{k=1,2}G_{2n-1,2k}∂_s\qty(α^{(k)}(s)-α_0^{(k)}(s)).
|
||||
\end{equation}
|
||||
|
||||
This can be evaluated using the exponential expansions and yields
|
||||
yet another sum of exponentials. The steady state flow can then be
|
||||
found be setting all exponentials to zero.
|
||||
364
src/hops_tweak.tex
Normal file
364
src/hops_tweak.tex
Normal file
|
|
@ -0,0 +1,364 @@
|
|||
\chapter{Some Notes on HOPS}
|
||||
\section{Normalized HOPS}%
|
||||
\label{sec:norm}
|
||||
|
||||
We introduce full HOPS vector \(Ψ = \qty(ψ, φ)\) which can be
|
||||
decomposed into the zeroth hierarchy order state \(ψ\) and the
|
||||
non-zero order states \(φ\).
|
||||
|
||||
The HOPS equations can then be written in an abstract manner as
|
||||
\begin{equation}
|
||||
\label{eq:HOPS}
|
||||
\begin{aligned}
|
||||
\dot{ψ} &= F(ψ, φ), & \dot{φ} &= G(ψ, φ),
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
where \(c\cdot F(ψ, φ) = F(c\cdot ψ, c\cdot φ)\) and
|
||||
\(c\cdot G(ψ, φ) = G(c\cdot ψ, c\cdot φ)\) for \(c\in\CC\)
|
||||
|
||||
|
||||
The goal is to transform \(ψ \rightarrow \tilde{ψ}\) so that
|
||||
\begin{equation}
|
||||
\label{eq:goal}
|
||||
\norm{\tilde{ψ}} = 1
|
||||
\end{equation}
|
||||
in a numerically stable manner.
|
||||
|
||||
Introducing the definitions \(\tilde{ψ} = \eu^{f(t)}ψ\) and
|
||||
\(\tilde{φ} = \eu^{f(t)}φ\) with an
|
||||
arbitrary\(f\colon \RR \rightarrow \CC\) we can begin to calculate
|
||||
\begin{equation}
|
||||
\label{eq:normdgl}
|
||||
∂_t\norm{\tilde{ψ}}^2 = \tilde{ψ}^† \qty(\dot{f}\tilde{ψ} +
|
||||
F(\tilde{ψ}, \tilde{φ})) + \cc = \dot{f} \abs{\tilde{ψ}}^2 +
|
||||
\tilde{ψ}^†F(\tilde{ψ}, \tilde{φ}) + \cc.
|
||||
\end{equation}
|
||||
|
||||
We would now like to obtain \(∂_t\norm{\tilde{ψ}}^2 = 0\) as well as
|
||||
\(\dot{f} > 0\) for \(\norm{\tilde{ψ}} < 0\) and \(\dot{f} < 0\) for
|
||||
\(\norm{\tilde{ψ}} > 0\), so that \(\norm{\tilde{ψ}} = 1\) becomes a
|
||||
stable fix-point.
|
||||
|
||||
Observing \cref{eq:normdgl}, we conclude that our goals can be
|
||||
achieved by demanding
|
||||
\begin{equation}
|
||||
\label{eq:fdgl}
|
||||
\dot{f} = \frac{\tilde{ψ}^†F(\tilde{ψ},
|
||||
\tilde{φ})}{\norm{\tilde{ψ}}^2} + g\qty(\norm{ψ}^2).
|
||||
\end{equation}
|
||||
The first summand on its own would lead to \(∂_t\norm{\tilde{ψ}}^2 =
|
||||
0\) norm conservation. The latter of our goals may be achieved by
|
||||
choosing \(g(x) = \qty(1-x)\).
|
||||
|
||||
These choices lead to an altered HOPS equation
|
||||
\begin{equation}
|
||||
\label{eq:normedhops}
|
||||
\dot{\tilde{Ψ}} = \qty[\frac{\tilde{ψ}^†F(\tilde{ψ},
|
||||
\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,
|
||||
\(H_B\nth = ∑_λω_λ\nth a_λ^{(n),†}a_λ\nth\),
|
||||
\(B_n=∑_{λ} g_λ\nth a_λ\nth\) 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}G_μ\nth\eu^{-W_μ\nth (t-s)}\fdv{η^\ast_n(s)}
|
||||
\end{equation}
|
||||
and
|
||||
\begin{equation}
|
||||
\label{eq:dops_full}
|
||||
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[-\iu H_\sys + \vb{L}\cdot\vb{η}^\ast -
|
||||
∑_{n=1}^N∑_{μ=1}^{M_n}\kmat_{n,μ}W\nth_μ]ψ^\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}
|
||||
|
||||
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}
|
||||
\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. Interestingly the divisor in
|
||||
\cref{eq:steadynorm_nonlin} may vanish, leading to a breakdown of the
|
||||
estimate. An interpretation would be, that for very strong coupling or
|
||||
long bath correlation times the interaction \fixme{reference} with the
|
||||
bath diverges and the method fails. On the other hand, the estimate
|
||||
may simply be wrong and should account for the coupling to the lower
|
||||
orders as well.
|
||||
|
||||
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.
|
||||
Loading…
Add table
Reference in a new issue