mirror of
https://github.com/vale981/master-thesis-tex
synced 2025-03-04 17:21:37 -05:00
398 lines
15 KiB
TeX
398 lines
15 KiB
TeX
\documentclass[final]{beamer}
|
||
|
||
\pdfvariable suppressoptionalinfo 512\relax
|
||
|
||
% ====================
|
||
% Packages
|
||
% ====================
|
||
|
||
\usepackage[T1]{fontenc}
|
||
\usepackage{../hiromacros}
|
||
|
||
\usepackage{lmodern}
|
||
\usepackage[size=A0,orientation=portrait,scale=1.0]{beamerposter}
|
||
\usetheme{gemini}
|
||
\usecolortheme{mit}
|
||
\usepackage{graphicx}
|
||
\usepackage{booktabs}
|
||
\usepackage{tikz}
|
||
\usepackage{pgfplots}
|
||
\pgfplotsset{compat=1.14}
|
||
\usepackage{physics}
|
||
\usepackage{cleveref}
|
||
\usepackage[font=scriptsize]{caption}
|
||
\usepackage[list=true, font=scriptsize, labelformat=brace]{subcaption}
|
||
\graphicspath{{figs/}}
|
||
|
||
% Bibliographic Stuff
|
||
\usepackage[backend=biber, language=english, style=phys]{biblatex}
|
||
\addbibresource{references.bib}
|
||
|
||
\DeclareDocumentCommand\vectorbold{ s m
|
||
}{\IfBooleanTF{#1}{\symbfit{#2}}{\symbf{#2}}} % Vector bold [star for
|
||
% Greek and italic Roman]
|
||
|
||
\makeatletter
|
||
\newcommand\thefontsize[1]{{#1 The current font size is: \f@size pt\par}}
|
||
\makeatother
|
||
|
||
% ====================
|
||
% Lengths
|
||
% ====================
|
||
|
||
% If you have N columns, choose \sepwidth and \colwidth such that
|
||
% (N+1)*\sepwidth + N*\colwidth = \paperwidth
|
||
\newlength{\sepwidth}
|
||
\newlength{\colwidth}
|
||
\setlength{\sepwidth}{0.066\paperwidth}
|
||
\setlength{\colwidth}{0.4\paperwidth}
|
||
|
||
\newcommand{\separatorcolumn}{\begin{column}{\sepwidth}\end{column}}
|
||
|
||
% ====================
|
||
% Title
|
||
% ====================
|
||
|
||
\title{Calculating Energy Flows in Strongly Coupled Open Quantum
|
||
Systems with HOPS}
|
||
|
||
\author{\underline{Valentin Boettcher}\inst{1}, Richard Hartmann\inst{1},
|
||
Konstantin Beyer\inst{1}, Walter Strunz\inst{1}}
|
||
|
||
\institute[shortinst]{\inst{1} Institute for Theoretical Physics, Dresden, Germany}
|
||
|
||
% ====================
|
||
% Footer (optional)
|
||
% ====================
|
||
|
||
\footercontent{
|
||
\href{https://tu-dresden.de/mn/physik/itp/tqo/die-professur}{https://tu-dresden.de/mn/physik/itp/tqo/die-professur} \hfill
|
||
Group for Theoretical Quantum Optics, TU-Dresden \hfill
|
||
\href{mailto:valentin.boettcher@tu-dresden.de}{valentin.boettcher@tu-dresden.de}}
|
||
% (can be left out to remove footer)
|
||
|
||
% ====================
|
||
% Logo (optional)
|
||
% ====================
|
||
|
||
% use this to include logos on the left and/or right side of the header:
|
||
\logoleft{\includegraphics[height=3cm]{Logo_TU_Dresden.pdf}}
|
||
|
||
% ====================
|
||
% Body
|
||
% ====================
|
||
|
||
\begin{document}
|
||
|
||
\begin{frame}[t]
|
||
\begin{columns}[t]
|
||
\separatorcolumn
|
||
|
||
\begin{column}{\colwidth}
|
||
\begin{block}{Premise}
|
||
\begin{itemize}
|
||
\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
|
||
important role \(\rightarrow\) must not be neglected in the
|
||
strong coupling regime
|
||
\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 (\(H_\sys(t)\))
|
||
coupled to \(N\) baths
|
||
\begin{equation}
|
||
\label{eq:generalmodel}
|
||
H(t) = H_\sys(t) + ∑_{n=1}^N \qty[L_n^†(t)B_n + \hc] + ∑_{n=1}^NH_B\nth ,
|
||
\end{equation}
|
||
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
|
||
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 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 hierarchy state \(\ket{Ψ}\in \hilb\otimes\mathcal{F}\)
|
||
\begin{equation}
|
||
\label{eq:fockhops}
|
||
\begin{aligned}
|
||
∂_t\ket{Ψ} &= \qty[
|
||
\begin{aligned}
|
||
-\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 \\
|
||
&\qquad+
|
||
\iu ∑_{n=1}^N∑_{μ=1}^{M_n} \sqrt{G_{n,μ}} \qty(b^†_{n,μ}L_n +
|
||
b_{n,μ}L^†_n)
|
||
\end{aligned}
|
||
] \ket{Ψ}.
|
||
\end{aligned}
|
||
\end{equation}
|
||
\item truncating hierarchy depth \(\kmat\) in \cref{eq:fockhops}
|
||
yields numeric method
|
||
|
||
\item finite temperature \(\rightarrow\) substitute
|
||
\(B(t)\rightarrow B(t)+ξ(t)\)
|
||
\item \(\exists\) nonlinear method which improves convergence drastically
|
||
\end{itemize}
|
||
|
||
See~\cite{Hartmann2017Dec} for details.
|
||
\end{block}
|
||
|
||
\begin{alertblock}{Bath Observables}
|
||
\begin{itemize}
|
||
\item \cref{eq:nmqsd,eq:fockhops} \(\implies\) correspondence
|
||
\(B^n(t) \leftrightarrow ψ^\kmat\) {\tiny(\(\abs{\kmat}=n\))}
|
||
|
||
\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.
|
||
|
||
\begin{description}
|
||
\item[Bath Energy Flow]
|
||
\begin{equation}
|
||
\label{eq:heatflowdef}
|
||
\begin{aligned}
|
||
J &= - \dv{\ev{H_\bath}}{t} = \ev{L(t)^†∂_t B(t) + L(t)∂_t
|
||
B^†(t)}_\inter \\
|
||
&=-\i \mathcal{M}_{η^\ast}\bra{\psi(η,
|
||
t)}L(t)^†\dot{D}_t\ket{\psi(η^\ast,t)} + \cc\\
|
||
&= - ∑_\mu\sqrt{G_\mu}W_\mu
|
||
\mathcal{M}_{η^\ast}\bra{\psi^{(0)}(η,t)}L(t)^†\ket{\psi^{\vb{e}_\mu}(η^\ast,t)} + \cc
|
||
\end{aligned}
|
||
\end{equation}
|
||
The expectation value of bath energy flow is connected to the first
|
||
level hierarchy states.
|
||
\item[Interaction Energy]
|
||
A similar expression exists for the expectation value of the
|
||
interaction energy.
|
||
\begin{equation}
|
||
\label{eq:intexp}
|
||
\begin{aligned}
|
||
\ev{H_\inter} &=-\i \mathcal{M}_{η^\ast}\bra{\psi(η,
|
||
t)}L(t)^†D_t\ket{\psi(η^\ast,t)} + \cc \\
|
||
&= ∑_\mu\sqrt{G_\mu}
|
||
\mathcal{M}_{η^\ast}\bra{\psi^{(0)}(η,t)}L(t)^†\ket{\psi^{\vb{e}_\mu}(η^\ast,t)} + \cc.
|
||
\end{aligned}
|
||
\end{equation}
|
||
\end{description}
|
||
|
||
This result allows us (in principle) to calculate the energy flow in
|
||
\textbf{arbitrarily driven systems} for a \textbf{wide temperature
|
||
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
|
||
\begin{equation}
|
||
\label{eq:hamiltonian}
|
||
\begin{aligned}
|
||
H = ∑_{i\in\qty{1,2}} &\qty[H^{(i)}_O + q_iB^{(i)} +
|
||
H_B^{(i)}]\\
|
||
&\quad+ \frac{γ}{4}(q_1-q_2)^2
|
||
\end{aligned}
|
||
\end{equation}
|
||
where \(H_O^{(i)}= \frac{Ω_i}{4}\qty(p_i^2+q_i^2)\)
|
||
\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},
|
||
\(α(τ)=η (1+\iu ω_cτ)^{-(2)}\) with
|
||
\tval{analytic/cutoff_freq}, \tval{analytic/bcf_zero}
|
||
\end{itemize}
|
||
\end{column}
|
||
\begin{column}{.5\colwidth}
|
||
\begin{figure}[H]
|
||
\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 show that the
|
||
analytical and numerical results are compatible.}
|
||
\end{figure}
|
||
\end{column}
|
||
\end{columns}
|
||
\end{block}
|
||
\begin{block}{Possible Applications}
|
||
\begin{itemize}
|
||
\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
|
||
\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}{Spin-Boson like Model and BCF Dependence}
|
||
\begin{itemize}
|
||
\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)^{-(2)}
|
||
\end{equation}
|
||
\item memory time \(\sim 1/ω_c\) has qualitative influence on the
|
||
bath energy flow
|
||
\end{itemize}
|
||
\begin{figure}[H]
|
||
\centering
|
||
\begin{subfigure}[t]{.49\columnwidth}
|
||
\plot{one_bath/omega_interaction}
|
||
\caption{\label{fig:omega_ints}\tval{one_bath/omega_bcf_str}}
|
||
\end{subfigure}
|
||
\begin{subfigure}[t]{.49\columnwidth}
|
||
\plot{one_bath/delta_interaction}
|
||
\caption{\tval{one_bath/delta_bcf_wc}}
|
||
\end{subfigure}
|
||
\caption{The interaction energy expectation value for different
|
||
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
|
||
\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
|
||
\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}, 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 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 \(L(τ) =
|
||
\sin^2(\frac{Δ}{2} τ)σ_x\)
|
||
\item Question: How much energy can be extracted from a system
|
||
connected to a single bath? (Ergotropy)
|
||
\begin{itemize}
|
||
\item Answer: less than
|
||
\(ΔE_{\mathrm{max}}=\frac{1}{β}\qrelent{ρ}{ρ_β}\)
|
||
\end{itemize}
|
||
\end{itemize}
|
||
\begin{figure}[H]
|
||
\centering
|
||
\begin{subfigure}[t]{.49\linewidth}
|
||
\plot{modcoup/omegas_total}
|
||
\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 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/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}{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}
|
||
|
||
\separatorcolumn
|
||
\end{columns}
|
||
\end{frame}
|
||
|
||
\end{document}
|