From 77424848dd3b421a6ec143ce4e621fd92743af3b Mon Sep 17 00:00:00 2001 From: Valentin Boettcher Date: Fri, 2 Sep 2022 16:20:02 +0200 Subject: [PATCH] move initial slip to the results section --- src/flow.tex | 120 ------------------------------------------- src/num_results.tex | 121 ++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 121 insertions(+), 120 deletions(-) diff --git a/src/flow.tex b/src/flow.tex index 88b2882..dac1f71 100644 --- a/src/flow.tex +++ b/src/flow.tex @@ -752,123 +752,3 @@ For the total power we find \end{equation} which can be evaluated as in \cref{sec:intener} by replacing \(L(t)\) with \(\dot{L}(t)\). - -\section{Pure Dephasing: The initial Slip} -\label{sec:pure_deph} -As seen in \fixme{include plots}, the short time behavior of the bath -energy flow is dominated by characteristic peak at short -times. Because this peak occurs at very short time scales, it may in -part be explained by a simple calculation which neglects the system -dynamics, setting \(H_\sys=0\). - -We solve the model with the Hamiltonian (Schr\"odinger picture) -\begin{equation} - \label{eq:puredeph} - H = L^†(t) B + L(t) B^† + H_\bath -\end{equation} -with \(L(t)=L(t)^†\), \([L(t), L(s)] = 0\;\forall t,s\) (so that -Heisenberg Hamiltonian matches \cref{eq:puredeph}) and \(B,H_\bath\) -as in \cref{eq:bop}. - -Because \([L,H]=0\) we can immediately solve \(L_H(t)=L_S(t)\), where -the subscript signify the Heisenberg and Schr\"odinger pictures -respectively. The Heisenberg equations for the \(a_λ\) yield -\begin{equation} - \label{eq:alapuredeph} - a_λ(t) = a_λ(0) \eu^{-\iu ω_λ t} - \iu g_λ^\ast∫_0^t\dd{s} L(s) - \eu^{-\iu ω_λ (t-s)}. -\end{equation} - -This allows us to calculate -\begin{equation} - \label{eq:pureflow} - \dot{H}_\bath = - ∑_λ g_λ L(t) \qty[∂_t a_λ(0) \eu^{\iu ω_λ t} - \iu - g_λ^\ast∫_0^t\dd{s} L(s) ∂_t \eu^{-\iu ω_λ (t-s)}] + \hc, -\end{equation} -which gives with a state of the form \(ρ=\ketbra{ψ} \otimes ρ_β\) -(\(ρ_β\) being a thermal state) -\begin{equation} - \label{eq:pureflowexpectation} - \ev{\dot{H}_\bath } = -2 ∫_0^t\dd{s}\ev{L(t)L(s)} \Im[\dot{α}(t-s)]. -\end{equation} - -For time independent \(L\) this becomes -\begin{equation} - \label{eq:pureflowtimeindep} - \ev{\dot{H}_\bath } = 2 \ev{L^2} \Im[\dot{α}(t)]. -\end{equation} - -The proportionality to the imaginary BCF \(α\) does explain the -initial peak in the bath energy flow. The imaginary part of the BCF is -zero for \(t=0\) and then usually features a peak at rather short -times (assuming finite correlation times). For the ohmic BCF used -here, this feature is very prominent. -\fixme{insert graph} - -Interestingly, \cref{eq:pureflowexpectation} does not contain any -reference to the temperature of the bath. Therefore, the bath energy -can only surpass its initial value in this model, as the dynamics -match the zero temperature case in which the bath has minimal energy -in the initial state. A thermodynamically useful model should -therefore feature an significant system dynamics that do not commute -with the interaction or fast modulation so that the Hamiltonian does -not commute with itself at different times. The latter may induce -deviations from the pure-dephasing behavior at very short time scales -and thus be useful for finite power output. \fixme{here the plot with - energy extraction would be good.} Coupling that is not self-adjoint -\fixme{plot} may also have this effect, but may be harder to -physically motivate. For the spin-boson system it is the result of the -random wave approximation, which however does not imply weak -coupling~\cite{Irish2007Oct}. - -For completeness, the interaction energy is given by -\begin{equation} - \label{eq:pureinter} - H_\inter = L(t)\qty[∑_λg_λ\qty(a_λ(0)\eu^{-\i ω_λ t} - \i - g^\ast_λ∫_0^t\dd{s} L(s) \eu^{\i ω_λ (t-s)})] + \hc, -\end{equation} -yielding -\begin{equation} - \label{eq:pureinterexp} - \ev{H_\inter} = 2 ∫_0^t\dd{s}\ev{L(t)L(s)} \Im[α(t-s)]. -\end{equation} -\fixme{plots} - -For time independent coupling we have -\begin{equation} - \label{eq:pureinterexp_timeidp} - \ev{H_\inter} = -2 \ev{L^2} \Im[α(t)] = -Δ\ev{H_{\bath}}. -\end{equation} - -It may be useful to normalize the BCF based on \cref{eq:pureinterexp}, -so that the pure interaction energy build-up in the initial slip is -canceled. To make the normalization independent of \(L(t)\), we choose -the normalization to be -\begin{equation} - \label{eq:bcfnorm} - \begin{aligned} - \mathcal{N} &= 2 \abs{\frac{\max_t\norm{L(t)L^\dag(t)+\hc}}{\max_t{\norm{H(t)}}} ∫_0^∞ \Im[α_u(τ)]\dd{τ}}\\ - α(τ) &= α_u(τ)/\mathcal{N}, - \end{aligned} -\end{equation} -where \(α_u\) is some unnormalized BCF. This normalization has the -useful property, that it neutralizes any scaling in \(L\). Note that -here the convention in which \(α\) is dimensionless is used. - -% this is not true -% imaginary part becomes proportional to the Dirac delta in the limit -% where typical cutoff frequency \(ω_c\rightarrow ∞\). The integral over -% the real part of \(α\) always gives zero if the spectral density obeys -% \(J(0) = 0\) and tends to exhibit fast oscillations and fast decay in -% the large-cutoff limit. For weak coupling, it may therefore be -% neglected. This constitutes the Markov limit mentioned in -% \cite{Strunz2001Habil}. - -The Ohmic-type BCF is -\begin{equation} - \label{eq:normohmic} - α(τ)=\frac{ω_c s }{ (\max_t\norm{H})(1+\iu ω_c τ)^{s+1}}, -\end{equation} -in this normalization. Note however, that the norm of the Hamiltonian -is assumed to be unity in the simulations referred to in this -thesis. \fixme{maybe change} diff --git a/src/num_results.tex b/src/num_results.tex index a5d8077..e134318 100644 --- a/src/num_results.tex +++ b/src/num_results.tex @@ -394,6 +394,127 @@ Hamiltonians. Because the NMQSD and also HOPS are largely agnostic of these factors, we may safely assume that the results of the comparison will be similar to the ones presented here. +\subsection{Pure Dephasing: The initial Slip} +\label{sec:pure_deph} +As seen in \cref{fig:comp_finite_t,fig:comp_zero_t,fig:comp_two_bath}, +the short time behavior of the bath energy flow is dominated by +characteristic peak at short times. Because this peak occurs at very +short time scales, it may in part be explained by a simple calculation +which neglects the system dynamics, setting \(H_\sys=0\). + +We solve the model with the Hamiltonian (Schr\"odinger picture) +\begin{equation} + \label{eq:puredeph} + H = L^†(t) B + L(t) B^† + H_\bath +\end{equation} +with \(L(t)=L(t)^†\), \([L(t), L(s)] = 0\;\forall t,s\) (so that +Heisenberg Hamiltonian matches \cref{eq:puredeph}) and \(B,H_\bath\) +as in \cref{eq:bop}. + +Because \([L,H]=0\) we can immediately solve \(L_H(t)=L_S(t)\), where +the subscript signify the Heisenberg and Schr\"odinger pictures +respectively. The Heisenberg equations for the \(a_λ\) yield +\begin{equation} + \label{eq:alapuredeph} + a_λ(t) = a_λ(0) \eu^{-\iu ω_λ t} - \iu g_λ^\ast∫_0^t\dd{s} L(s) + \eu^{-\iu ω_λ (t-s)}. +\end{equation} + +This allows us to calculate +\begin{equation} + \label{eq:pureflow} + \dot{H}_\bath = - ∑_λ g_λ L(t) \qty[∂_t a_λ(0) \eu^{\iu ω_λ t} - \iu + g_λ^\ast∫_0^t\dd{s} L(s) ∂_t \eu^{-\iu ω_λ (t-s)}] + \hc, +\end{equation} +which gives with a state of the form \(ρ=\ketbra{ψ} \otimes ρ_β\) +(\(ρ_β\) being a thermal state) +\begin{equation} + \label{eq:pureflowexpectation} + \ev{\dot{H}_\bath } = -2 ∫_0^t\dd{s}\ev{L(t)L(s)} \Im[\dot{α}(t-s)]. +\end{equation} + +For time independent \(L\) this becomes +\begin{equation} + \label{eq:pureflowtimeindep} + \ev{\dot{H}_\bath } = 2 \ev{L^2} \Im[\dot{α}(t)]. +\end{equation} + +The proportionality to the imaginary BCF \(α\) does explain the +initial peak in the bath energy flow. The imaginary part of the BCF is +zero for \(t=0\) and then usually features a peak at rather short +times (assuming finite correlation times). For the ohmic BCF used +here, this feature is very prominent. +\fixme{insert graph} + +Interestingly, \cref{eq:pureflowexpectation} does not contain any +reference to the temperature of the bath. Therefore, the bath energy +can only surpass its initial value in this model, as the dynamics +match the zero temperature case in which the bath has minimal energy +in the initial state. A thermodynamically useful model should +therefore feature an significant system dynamics that do not commute +with the interaction or fast modulation so that the Hamiltonian does +not commute with itself at different times. The latter may induce +deviations from the pure-dephasing behavior at very short time scales +and thus be useful for finite power output. \fixme{here the plot with + energy extraction would be good.} Coupling that is not self-adjoint +\fixme{plot} may also have this effect, but may be harder to +physically motivate. For the spin-boson system it is the result of the +random wave approximation, which however does not imply weak +coupling~\cite{Irish2007Oct}. + +For completeness, the interaction energy is given by +\begin{equation} + \label{eq:pureinter} + H_\inter = L(t)\qty[∑_λg_λ\qty(a_λ(0)\eu^{-\i ω_λ t} - \i + g^\ast_λ∫_0^t\dd{s} L(s) \eu^{\i ω_λ (t-s)})] + \hc, +\end{equation} +yielding +\begin{equation} + \label{eq:pureinterexp} + \ev{H_\inter} = 2 ∫_0^t\dd{s}\ev{L(t)L(s)} \Im[α(t-s)]. +\end{equation} +\fixme{plots} + +For time independent coupling we have +\begin{equation} + \label{eq:pureinterexp_timeidp} + \ev{H_\inter} = -2 \ev{L^2} \Im[α(t)] = -Δ\ev{H_{\bath}}. +\end{equation} + +It may be useful to normalize the BCF based on \cref{eq:pureinterexp}, +so that the pure interaction energy build-up in the initial slip is +canceled. To make the normalization independent of \(L(t)\), we choose +the normalization to be +\begin{equation} + \label{eq:bcfnorm} + \begin{aligned} + \mathcal{N} &= 2 \abs{\frac{\max_t\norm{L(t)L^\dag(t)+\hc}}{\max_t{\norm{H(t)}}} ∫_0^∞ \Im[α_u(τ)]\dd{τ}}\\ + α(τ) &= α_u(τ)/\mathcal{N}, + \end{aligned} +\end{equation} +where \(α_u\) is some unnormalized BCF. This normalization has the +useful property, that it neutralizes any scaling in \(L\). Note that +here the convention in which \(α\) is dimensionless is used. + +% this is not true +% imaginary part becomes proportional to the Dirac delta in the limit +% where typical cutoff frequency \(ω_c\rightarrow ∞\). The integral over +% the real part of \(α\) always gives zero if the spectral density obeys +% \(J(0) = 0\) and tends to exhibit fast oscillations and fast decay in +% the large-cutoff limit. For weak coupling, it may therefore be +% neglected. This constitutes the Markov limit mentioned in +% \cite{Strunz2001Habil}. + +The Ohmic-type BCF is +\begin{equation} + \label{eq:normohmic} + α(τ)=\frac{ω_c s }{ (\max_t\norm{H})(1+\iu ω_c τ)^{s+1}}, +\end{equation} +in this normalization. Note however, that the norm of the Hamiltonian +is assumed to be unity in the simulations referred to in this +thesis. \fixme{maybe change} + + \section{Precision Simulations for a System without Analytical Solution} \label{sec:prec_sim}