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add anti zeno, final cleanup

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Valentin Boettcher 2022-07-13 15:28:12 +02:00
parent 1ca212af38
commit 129f181618
46 changed files with 115273 additions and 65121 deletions
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%% Creator: Matplotlib, PGF backend
%%
%% To include the figure in your LaTeX document, write
%% \input{<filename>.pgf}
%%
%% Make sure the required packages are loaded in your preamble
%% \usepackage{pgf}
%%
%% Also ensure that all the required font packages are loaded; for instance,
%% the lmodern package is sometimes necessary when using math font.
%% \usepackage{lmodern}
%%
%% Figures using additional raster images can only be included by \input if
%% they are in the same directory as the main LaTeX file. For loading figures
%% from other directories you can use the `import` package
%% \usepackage{import}
%%
%% and then include the figures with
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%%
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poster/figs/one_bath/values/omega_bcf_str.tex
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\(α(0)=0.3\)

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poster/poster.tex
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\documentclass[draft]{beamer}
\documentclass[final]{beamer}
\pdfvariable suppressoptionalinfo 512\relax
@ -20,7 +20,8 @@
\pgfplotsset{compat=1.14}
\usepackage{physics}
\usepackage{cleveref}
\usepackage[list=true, font=footnotesize, labelformat=brace]{subcaption}
\usepackage[font=scriptsize]{caption}
\usepackage[list=true, font=scriptsize, labelformat=brace]{subcaption}
\graphicspath{{figs/}}
% Bibliographic Stuff
@ -90,24 +91,23 @@
\begin{column}{\colwidth}
\begin{block}{Premise}
\begin{itemize}
\item Application of thermodynamic notions to strongly coupled and
non-Markovian quantum systems is non-trivial
\item application of thermodynamic notions to strongly coupled and
non-Markovian open quantum systems is non-trivial
(\cite{Rivas2019Oct,Kato2016Dec,Strasberg2021Aug,Talkner2020Oct} and many
more)
\item Dynamics of bath and interaction hamiltonians plays an
\item dynamics of bath and interaction Hamiltonians plays an
important role \(\rightarrow\) must not be neglected in the
strong coupling regime
\item The ``Hierarchy of Pure States''
(HOPS~\cite{Hartmann2017Dec,Diosi1998Mar}) gives us the ability
to simulate non-Markovian and strongly coupled open quantum
systems exactly in a scalable way.
\item Because HOPS simulates global dynamics \(\rightarrow\) gives
access to certain bath dynamics \emph{with no additional effort}
\item the ``Hierarchy of Pure States''
(HOPS~\cite{Hartmann2017Dec,Diosi1998Mar}) gives the us ability to
simulate open quantum systems \textbf{exactly}
\item because HOPS simulates global dynamics \(\rightarrow\) \textbf{gives
access to certain bath dynamics with no additional effort}
\end{itemize}
\end{block}
\begin{block}{NMQSD/HOPS}
\begin{itemize}
\item Consider the model of a general quantum system (\(H_\sys(t)\))
\item consider the model of a general quantum system (\(H_\sys(t)\))
coupled to \(N\) baths
\begin{equation}
\label{eq:generalmodel}
@ -116,18 +116,18 @@
with \(B_n=∑_{λ} g_λ\nth a_λ\nth\) and
\(H_B\nth=∑_λω_λ\nth \qty(b_λ\nth)^\dag b_λ\nth\)
\item Leads to \emph{stochastic} Non-Markovian
\item leads to \emph{stochastic} Non-Markovian
Quantum State Diffusion (NMQSD)
\begin{equation}
\label{eq:nmqsd}
∂_tψ_t(\vb{η}^\ast_t) = -\iu H(t) ψ_t(\vb{η}^\ast_t) +
\vb{L}\cdot \vb{η}^\ast_tψ_t(\vb{η}^\ast_t) - ∑_{n=1}^N L(t)_n^†∫_0^t\dd{s}α_n(t-s)\fdv{ψ_t(\vb{η}^\ast_t)}{η^\ast_n(s)}
\end{equation}
\item Reduced state of the system is recovered through
\item the reduced state of the system is recovered through
\(ρ=\mathcal{M}(ψ_t(\vb{η}^\ast_t)ψ_t^\dag(\vb{η}^\ast_t))\)
\item With \(α_n(τ)=∑_{\mu}^{M_n}G_μ\nth\eu^{-W_μ\nth τ}\)
we find for the Fock-space embedded hierarchy state \(\ket{Ψ}\)
\item with \(α_n(τ)=∑_{\mu}^{M_n}G_μ\nth\eu^{-W_μ\nth τ}\)
we find for the hierarchy state \(\ket{Ψ}\in \hilb\otimes\mathcal{F}\)
\begin{equation}
\label{eq:fockhops}
\begin{aligned}
@ -142,12 +142,11 @@
] \ket{Ψ}.
\end{aligned}
\end{equation}
\item Truncating hierarchy depth \(\kmat\) in \cref{eq:fockhops}
\item truncating hierarchy depth \(\kmat\) in \cref{eq:fockhops}
yields numeric method
\item Finite temperature \(\rightarrow\) substitute
\(B(t)\rightarrow B(t)+ξ(t)\), \(ξ\) being a suitable
Gaussian process
\item finite temperature \(\rightarrow\) substitute
\(B(t)\rightarrow B(t)+ξ(t)\)
\item \(\exists\) nonlinear method which improves convergence drastically
\end{itemize}
@ -156,14 +155,13 @@
\begin{alertblock}{Bath Observables}
\begin{itemize}
\item \Cref{eq:nmqsd,eq:fockhops} \(\implies\) correspondence
\item \cref{eq:nmqsd,eq:fockhops} \(\implies\) correspondence
\(B^n(t) \leftrightarrow ψ^\kmat\) {\tiny(\(\abs{\kmat}=n\))}
\item Can calulate observables of type
\item can calulate observables of type
\(O_\sys\otimes (B^a)^\dag B^b\) + time derivatives
\end{itemize}
We consider the zero temperature case with one bath in the linear
method.
@ -180,18 +178,11 @@
\mathcal{M}_{η^\ast}\bra{\psi^{(0)}(η,t)}L(t)^†\ket{\psi^{\vb{e}_\mu}(η^\ast,t)} + \cc
\end{aligned}
\end{equation}
\begin{itemize}
\item Expectation value of bath energy flow \(\leftrightarrow\)
first level hierarchy states in a transparent and easy to calculate
manner
\end{itemize}
The expectation value of bath energy flow is connected to the first
level hierarchy states.
\item[Interaction Energy]
\begin{itemize}
\item Similar expression exists the expectation value of the
interaction energy
\end{itemize}
A similar expression exists for the expectation value of the
interaction energy.
\begin{equation}
\label{eq:intexp}
\begin{aligned}
@ -203,16 +194,16 @@
\end{equation}
\end{description}
This result allows us to calculate the energy flow in
This result allows us (in principle) to calculate the energy flow in
\textbf{arbitrarily driven systems} for a \textbf{wide temperature
range} and with \textbf{arbitrary BCF}.
range} and with (sub-)Ohmic BCF.
\end{alertblock}
\begin{block}{Comparison to an Analytic Solution}
\begin{columns}
\begin{column}{.5\colwidth}
\begin{itemize}
\item Quantum Brownian Motion like Model
\item ``Quantum Brownian Motion'' like model
\begin{equation}
\label{eq:hamiltonian}
\begin{aligned}
@ -222,11 +213,11 @@
\end{aligned}
\end{equation}
where \(H_O^{(i)}= \frac{Ω_i}{4}\qty(p_i^2+q_i^2)\)
\item Exact solution via exponential expansion of the BCF in the
Heisenberg Picture \(\rightarrow\) easy access to bath energy
\item exact solution via exponential expansion of the BCF in the
Heisenberg picture \(\rightarrow\) easy access to bath energy
flow
\item Here \tval{analytic/omega}, \tval{analytic/gamma},
\(α(τ)=η (1+\iu ω_cτ)^{-(s+1)}\) with
\item here \tval{analytic/omega}, \tval{analytic/gamma},
\(α(τ)=η (1+\iu ω_cτ)^{-(2)}\) with
\tval{analytic/cutoff_freq}, \tval{analytic/bcf_zero}
\end{itemize}
\end{column}
@ -235,38 +226,37 @@
\centering
\plot{analytic/flow}
\caption{\label{fig:brownian}The bath energy flows \cref{eq:heatflowdef} for the
hot (lower line) and the cold (upper line) bath. The
analytical and numerical results are compatible.
(\tval{analytic/samples} samples)}
hot (lower line) and the cold (upper line) bath show that the
analytical and numerical results are compatible.}
\end{figure}
\end{column}
\end{columns}
\end{block}
\begin{block}{Possible Applications}
\begin{itemize}
\item Simulation of Thermal Quantum Machines
\item Convergence Criteria: Energy Conservation, Calculating the
\item \emph{Simulation of thermal quantum machines}
\item convergence criteria for HOPS: energy conservation, calculating the
same observable in multiple ways
\item Quantification of Entanglement of System and Bath (Fisher
Information of \(H_\inter\))
\item \ldots
\item quantification of entanglement of system and bath
\item testing results obtained from approximations
\end{itemize}
\end{block}
\begin{block}{Resources}
{\AtNextBibliography{\tiny} \printbibliography}
\end{block}
\end{column}
\separatorcolumn
\begin{column}{\colwidth}
\begin{block}{Simple Toy Model and BCF Dependence}
\begin{block}{Spin-Boson like Model and BCF Dependence}
\begin{itemize}
\item Spin-boson like Model coupled to a zero temperature bath
\item spin-boson like model coupled to a zero temperature bath
\begin{equation}
\label{eq:spinbos}
H_\sys= \frac12 σ_z,\, L=\frac12 σ_x,\, α(τ)=η (1+\iu ω_cτ)^{-(s+1)}
H_\sys= \frac12 σ_z,\, L=\frac12 σ_x,\, α(τ)=η (1+\iu ω_cτ)^{-(2)}
\end{equation}
\item Memory time \(\sim 1/ω_c\) has qualitative influence on the
\item memory time \(\sim 1/ω_c\) has qualitative influence on the
bath energy flow
\end{itemize}
\begin{figure}[H]
@ -280,42 +270,43 @@
\caption{\tval{one_bath/delta_bcf_wc}}
\end{subfigure}
\caption{The interaction energy expectation value for different
cutoff frequencies and coupling strengths. The dashed lines
are obtained using energy conservation while the solid lines
are the result of direct calculation. The percentages in the
legend tell how many points are compatible within one standard
deviation. The Statistical error estimate is smaller than the
line width. \(N=5\cdot 10^5\) trajectories have been used.}
cutoff frequencies and coupling strengths, where the dashed
lines are obtained using energy conservation while the solid
lines are the result of direct calculation. The percentages in
the legend tell how many points are compatible within one
standard deviation. The Statistical error estimate is smaller
than the line width. \(N=5\cdot 10^5\) trajectories have been
used.}
\end{figure}
\end{block}
\begin{block}{Initial Slip}
\begin{itemize}
\item For \emph{very} short times \(\rightarrow\) \(H_\sys\approx
0\), Origin of the \emph{``Initial Slip''} spike in
\item for \emph{very} short times \(\rightarrow\) \(H_\sys\approx
0\), origin of the \emph{``Initial Slip''} spike in
\cref{fig:brownian}:
\begin{equation}
\label{eq:purede}
\ev{\dot{H}_\bath } = -2 ∫_0^t\dd{s}\ev{L(t)L(s)} \Im[\dot{α}(t-s)].
\end{equation}
\item Determines ultra short-time shape of \emph{all} trajectories
\item determines ultra short-time shape of \emph{all} trajectories
\end{itemize}
\begin{figure}[H]
\centering
\begin{subfigure}[t]{.49\linewidth}
\plot{one_bath/initial_slip}
\caption{\label{fig:initslipconst}The bath energy flows for the same settings as in
\cref{fig:omega_ints}. The dashed lines correspond to \cref{eq:purede}.}
\cref{fig:omega_ints}, where the dashed lines correspond to \cref{eq:purede}.}
\end{subfigure}
\begin{subfigure}[t]{.49\linewidth}
\plot{modcoup/initial_slip_modcoup}
\caption{Same as \cref{fig:initslipconst}, but for modulated
coupling (``Smoothstep'' functions, see inset).}
coupling (``Smoothstep'' functions with smoothness \(s\), see inset).}
\end{subfigure}
\end{figure}
\end{block}
\begin{block}{Modulating the Coupling}
\begin{itemize}
\item Same model as above \cref{eq:spinbos}, but with \(L(τ) =
\item same model as above \cref{eq:spinbos}, but with \(L(τ) =
\sin^2(\frac{Δ}{2} τ)σ_x\)
\item Question: How much energy can be extracted from a system
connected to a single bath? (Ergotropy)
@ -331,28 +322,72 @@
\caption{\label{fig:omega_total}The total energy for
\tval{modcoup/omega_delta} and \tval{modcoup/omega_alpha}
but varying cutoff. Energy is normalized to the
ergotropy. The dashed vertical lines show where \(α(τ) = α(0)/300\).}
ergotropy. The dashed vertical lines illustrate the bath
memory time (\(|α(τ)| = α(0)/300\)).}
\end{subfigure}
% \begin{subfigure}[t]{.49\linewidth}
% \plot{modcoup/flow_interaction_overview}
% \caption{Same situation as in \cref{fig:omega_total}. All
% quantities are normalized to the \(ω_c=1\)
% case.}
% \end{subfigure}
\begin{subfigure}[t]{.49\linewidth}
\plot{modcoup/flow_interaction_overview}
\caption{Same situation as in \cref{fig:omega_total}. All
quantities are normalized to the \(ω_c=1\)
case.}
\end{subfigure}
\begin{subfigure}[t]{.49\linewidth}
\plot{modcoup/flow_interaction_overview}
\caption{Maximum power for \(10\) periods and different
modulations \(Δ\).}
\end{subfigure}
\begin{subfigure}[t]{.49\linewidth}
\plot{modcoup/flow_interaction_overview}
\caption{Maximum power for \(10\) periods and different
coupling strengths \(α(0)\).}
\plot{modcoup/delta_dependence}
\caption{Maximum one-shot power for \(10\) periods and
different modulation frequencies \(Δ\) and coupling
strengths.}
\end{subfigure}
\end{figure}
\end{block}
\begin{block}{Resources}
{\AtNextBibliography{\tiny} \printbibliography}
\begin{block}{Continuously Coupled Engine (Preliminary)}
\begin{itemize}
\item qubit coupled to two baths of different
temperatures (\(T_c, T_h\))
\begin{equation}
\label{eq:antizenomodel}
H_\sys= \frac12 \qty[ω_0 + γ Δ\sin(Δ t)]σ_z,\, L_{c,h}=\frac12 σ_x
\end{equation}
{\tiny \tval{anti_zeno/delta}, \tval{anti_zeno/gamma}, \tval{anti_zeno/omega_alpha},
\tval{anti_zeno/omega_zero}, \tval{anti_zeno/tc}, \tval{anti_zeno/th}}
\item system Hamiltonian modulated + baths periodically
decoupled (cooldown) and ``reset''
\end{itemize}
\begin{figure}[H]
\centering
\begin{subfigure}[t]{.49\linewidth}
\plot{anti_zeno/sd_setup}
\caption{The spectral densities of the baths, where the vertical
lines show where \(ω=ω_0 \pm Δ\). The overlap of the filter
for \(n\) modulation periods is crucial for the ``anti-zeno''
effect~\cite{Mukherjee2020Jan}.}
\end{subfigure}
% \begin{subfigure}[t]{.49\linewidth}
% \plot{modcoup/flow_interaction_overview}
% \caption{Same situation as in \cref{fig:omega_total}. All
% quantities are normalized to the \(ω_c=1\)
% case.}
% \end{subfigure}
\begin{subfigure}[t]{.49\linewidth}
\plot{anti_zeno/modulation_setup}
\caption{Setup of the system and coupling modulation for the
case with ``cooldown''. The units are arbitrary. The
initialization period and two cycles are shown.}
\end{subfigure}
\begin{subfigure}[t]{.49\linewidth}
\plot{anti_zeno/anti_zeno_with_cool}
\caption{\label{fig:cont_coup} The total energy change after the initialization
period, where vertical lines show the points at which the
times for the calculation of the power are taken. The mean
power obtained is \tval{anti_zeno/power_with_cool}.}.
\end{subfigure}
\begin{subfigure}[t]{.49\linewidth}
\plot{anti_zeno/anti_zeno_without_cool}
\caption{Same as in \cref{fig:cont_coup} but without
cooldown. The mean power obtained is
\tval{anti_zeno/power_without_cool}. The difference is not
significant yet.}.
\end{subfigure}
\end{figure}
\end{block}
\end{column}

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poster/values/anti_zeno/delta.tex Normal file
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\(Δ=11\)

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poster/values/anti_zeno/gamma.tex Normal file
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\(γ=0.5\)

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poster/values/anti_zeno/omega_alpha.tex Normal file
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\(α(0)=1.0\)

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poster/values/anti_zeno/omega_zero.tex Normal file
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\(ω_0=20\)

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poster/values/anti_zeno/power_with_cool.tex Normal file
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\(P=-0.068\pm 0.010\)

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poster/values/anti_zeno/power_without_cool.tex Normal file
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\(P=-0.058\pm 0.014\)

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poster/values/anti_zeno/tc.tex Normal file
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\(T_c=8\)

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poster/values/anti_zeno/th.tex Normal file
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\(T_h=40\)