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move initial slip to the results section

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120
src/flow.tex
View file

@ -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}

121
src/num_results.tex
View file

@ -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}