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398 lines
15 KiB
TeX
398 lines
15 KiB
TeX
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% ====================
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% Title
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% ====================
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\title{Calculating Energy Flows in Strongly Coupled Open Quantum
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Systems with HOPS}
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\author{\underline{Valentin Boettcher}\inst{1}, Richard Hartmann\inst{1},
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Konstantin Beyer\inst{1}, Walter Strunz\inst{1}}
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\institute[shortinst]{\inst{1} Institute for Theoretical Physics, Dresden, Germany}
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% ====================
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\footercontent{
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\href{https://tu-dresden.de/mn/physik/itp/tqo/die-professur}{https://tu-dresden.de/mn/physik/itp/tqo/die-professur} \hfill
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Group for Theoretical Quantum Optics, TU-Dresden \hfill
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\href{mailto:valentin.boettcher@tu-dresden.de}{valentin.boettcher@tu-dresden.de}}
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\logoleft{\includegraphics[height=3cm]{Logo_TU_Dresden.pdf}}
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% ====================
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% Body
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% ====================
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\begin{document}
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\begin{frame}[t]
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\begin{columns}[t]
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\separatorcolumn
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\begin{column}{\colwidth}
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\begin{block}{Premise}
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\begin{itemize}
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\item application of thermodynamic notions to strongly coupled and
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non-Markovian open quantum systems is non-trivial
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(\cite{Rivas2019Oct,Kato2016Dec,Strasberg2021Aug,Talkner2020Oct} and many
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more)
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\item dynamics of bath and interaction Hamiltonians plays an
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important role \(\rightarrow\) must not be neglected in the
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strong coupling regime
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\item the ``Hierarchy of Pure States''
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(HOPS~\cite{Hartmann2017Dec,Diosi1998Mar}) gives the us ability to
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simulate open quantum systems \textbf{exactly}
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\item because HOPS simulates global dynamics \(\rightarrow\) \textbf{gives
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access to certain bath dynamics with no additional effort}
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\end{itemize}
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\end{block}
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\begin{block}{NMQSD/HOPS}
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\begin{itemize}
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\item consider the model of a general quantum system (\(H_\sys(t)\))
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coupled to \(N\) baths
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\begin{equation}
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\label{eq:generalmodel}
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H(t) = H_\sys(t) + ∑_{n=1}^N \qty[L_n^†(t)B_n + \hc] + ∑_{n=1}^NH_B\nth ,
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\end{equation}
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with \(B_n=∑_{λ} g_λ\nth a_λ\nth\) and
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\(H_B\nth=∑_λω_λ\nth \qty(b_λ\nth)^\dag b_λ\nth\)
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\item leads to \emph{stochastic} Non-Markovian
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Quantum State Diffusion (NMQSD)
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\begin{equation}
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\label{eq:nmqsd}
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∂_tψ_t(\vb{η}^\ast_t) = -\iu H(t) ψ_t(\vb{η}^\ast_t) +
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\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)}
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\end{equation}
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\item the reduced state of the system is recovered through
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\(ρ=\mathcal{M}(ψ_t(\vb{η}^\ast_t)ψ_t^\dag(\vb{η}^\ast_t))\)
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\item with \(α_n(τ)=∑_{\mu}^{M_n}G_μ\nth\eu^{-W_μ\nth τ}\)
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we find for the hierarchy state \(\ket{Ψ}\in \hilb\otimes\mathcal{F}\)
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\begin{equation}
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\label{eq:fockhops}
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\begin{aligned}
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∂_t\ket{Ψ} &= \qty[
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\begin{aligned}
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-\iu H_\sys + \vb{L}\cdot\vb{η}^\ast &-
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∑_{n=1}^N∑_{μ=1}^{M_n}b_{n,μ}^\dag b_{n,μ} W\nth_μ \\
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&\qquad+
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\iu ∑_{n=1}^N∑_{μ=1}^{M_n} \sqrt{G_{n,μ}} \qty(b^†_{n,μ}L_n +
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b_{n,μ}L^†_n)
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\end{aligned}
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] \ket{Ψ}.
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\end{aligned}
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\end{equation}
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\item truncating hierarchy depth \(\kmat\) in \cref{eq:fockhops}
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yields numeric method
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\item finite temperature \(\rightarrow\) substitute
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\(B(t)\rightarrow B(t)+ξ(t)\)
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\item \(\exists\) nonlinear method which improves convergence drastically
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\end{itemize}
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See~\cite{Hartmann2017Dec} for details.
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\end{block}
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\begin{alertblock}{Bath Observables}
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\begin{itemize}
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\item \cref{eq:nmqsd,eq:fockhops} \(\implies\) correspondence
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\(B^n(t) \leftrightarrow ψ^\kmat\) {\tiny(\(\abs{\kmat}=n\))}
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\item can calulate observables of type
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\(O_\sys\otimes (B^a)^\dag B^b\) + time derivatives
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\end{itemize}
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We consider the zero temperature case with one bath in the linear
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method.
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\begin{description}
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\item[Bath Energy Flow]
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\begin{equation}
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\label{eq:heatflowdef}
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\begin{aligned}
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J &= - \dv{\ev{H_\bath}}{t} = \ev{L(t)^†∂_t B(t) + L(t)∂_t
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B^†(t)}_\inter \\
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&=-\i \mathcal{M}_{η^\ast}\bra{\psi(η,
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t)}L(t)^†\dot{D}_t\ket{\psi(η^\ast,t)} + \cc\\
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&= - ∑_\mu\sqrt{G_\mu}W_\mu
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\mathcal{M}_{η^\ast}\bra{\psi^{(0)}(η,t)}L(t)^†\ket{\psi^{\vb{e}_\mu}(η^\ast,t)} + \cc
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\end{aligned}
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\end{equation}
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The expectation value of bath energy flow is connected to the first
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level hierarchy states.
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\item[Interaction Energy]
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A similar expression exists for the expectation value of the
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interaction energy.
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\begin{equation}
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\label{eq:intexp}
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\begin{aligned}
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\ev{H_\inter} &=-\i \mathcal{M}_{η^\ast}\bra{\psi(η,
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t)}L(t)^†D_t\ket{\psi(η^\ast,t)} + \cc \\
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&= ∑_\mu\sqrt{G_\mu}
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\mathcal{M}_{η^\ast}\bra{\psi^{(0)}(η,t)}L(t)^†\ket{\psi^{\vb{e}_\mu}(η^\ast,t)} + \cc.
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\end{aligned}
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\end{equation}
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\end{description}
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This result allows us (in principle) to calculate the energy flow in
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\textbf{arbitrarily driven systems} for a \textbf{wide temperature
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range} and with (sub-)Ohmic BCF.
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\end{alertblock}
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\begin{block}{Comparison to an Analytic Solution}
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\begin{columns}
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\begin{column}{.5\colwidth}
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\begin{itemize}
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\item ``Quantum Brownian Motion'' like model
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\begin{equation}
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\label{eq:hamiltonian}
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\begin{aligned}
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H = ∑_{i\in\qty{1,2}} &\qty[H^{(i)}_O + q_iB^{(i)} +
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H_B^{(i)}]\\
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&\quad+ \frac{γ}{4}(q_1-q_2)^2
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\end{aligned}
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\end{equation}
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where \(H_O^{(i)}= \frac{Ω_i}{4}\qty(p_i^2+q_i^2)\)
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\item exact solution via exponential expansion of the BCF in the
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Heisenberg picture \(\rightarrow\) easy access to bath energy
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flow
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\item here \tval{analytic/omega}, \tval{analytic/gamma},
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\(α(τ)=η (1+\iu ω_cτ)^{-(2)}\) with
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\tval{analytic/cutoff_freq}, \tval{analytic/bcf_zero}
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\end{itemize}
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\end{column}
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\begin{column}{.5\colwidth}
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\begin{figure}[H]
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\centering
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\plot{analytic/flow}
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\caption{\label{fig:brownian}The bath energy flows \cref{eq:heatflowdef} for the
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hot (lower line) and the cold (upper line) bath show that the
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analytical and numerical results are compatible.}
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\end{figure}
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\end{column}
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\end{columns}
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\end{block}
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\begin{block}{Possible Applications}
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\begin{itemize}
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\item \emph{Simulation of thermal quantum machines}
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\item convergence criteria for HOPS: energy conservation, calculating the
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same observable in multiple ways
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\item quantification of entanglement of system and bath
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\item testing results obtained from approximations
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\end{itemize}
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\end{block}
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\begin{block}{Resources}
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{\AtNextBibliography{\tiny} \printbibliography}
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\end{block}
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\end{column}
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\separatorcolumn
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\begin{column}{\colwidth}
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\begin{block}{Spin-Boson like Model and BCF Dependence}
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\begin{itemize}
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\item spin-boson like model coupled to a zero temperature bath
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\begin{equation}
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\label{eq:spinbos}
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H_\sys= \frac12 σ_z,\, L=\frac12 σ_x,\, α(τ)=η (1+\iu ω_cτ)^{-(2)}
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\end{equation}
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\item memory time \(\sim 1/ω_c\) has qualitative influence on the
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bath energy flow
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\end{itemize}
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\begin{figure}[H]
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\centering
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\begin{subfigure}[t]{.49\columnwidth}
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\plot{one_bath/omega_interaction}
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\caption{\label{fig:omega_ints}\tval{one_bath/omega_bcf_str}}
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\end{subfigure}
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\begin{subfigure}[t]{.49\columnwidth}
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\plot{one_bath/delta_interaction}
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\caption{\tval{one_bath/delta_bcf_wc}}
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\end{subfigure}
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\caption{The interaction energy expectation value for different
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cutoff frequencies and coupling strengths, where the dashed
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lines are obtained using energy conservation while the solid
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lines are the result of direct calculation. The percentages in
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the legend tell how many points are compatible within one
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standard deviation. The Statistical error estimate is smaller
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than the line width. \(N=5\cdot 10^5\) trajectories have been
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used.}
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\end{figure}
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\end{block}
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\begin{block}{Initial Slip}
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\begin{itemize}
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\item for \emph{very} short times \(\rightarrow\) \(H_\sys\approx
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0\), origin of the \emph{``Initial Slip''} spike in
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\cref{fig:brownian}:
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\begin{equation}
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\label{eq:purede}
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\ev{\dot{H}_\bath } = -2 ∫_0^t\dd{s}\ev{L(t)L(s)} \Im[\dot{α}(t-s)].
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\end{equation}
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\item determines ultra short-time shape of \emph{all} trajectories
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\end{itemize}
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\begin{figure}[H]
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\centering
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\begin{subfigure}[t]{.49\linewidth}
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\plot{one_bath/initial_slip}
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\caption{\label{fig:initslipconst}The bath energy flows for the same settings as in
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\cref{fig:omega_ints}, where the dashed lines correspond to \cref{eq:purede}.}
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\end{subfigure}
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\begin{subfigure}[t]{.49\linewidth}
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\plot{modcoup/initial_slip_modcoup}
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\caption{Same as \cref{fig:initslipconst}, but for modulated
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coupling (``Smoothstep'' functions with smoothness \(s\), see inset).}
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\end{subfigure}
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\end{figure}
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\end{block}
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\begin{block}{Modulating the Coupling}
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\begin{itemize}
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\item same model as above \cref{eq:spinbos}, but with \(L(τ) =
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\sin^2(\frac{Δ}{2} τ)σ_x\)
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\item Question: How much energy can be extracted from a system
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connected to a single bath? (Ergotropy)
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\begin{itemize}
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\item Answer: less than
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\(ΔE_{\mathrm{max}}=\frac{1}{β}\qrelent{ρ}{ρ_β}\)
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\end{itemize}
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\end{itemize}
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\begin{figure}[H]
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\centering
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\begin{subfigure}[t]{.49\linewidth}
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\plot{modcoup/omegas_total}
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\caption{\label{fig:omega_total}The total energy for
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\tval{modcoup/omega_delta} and \tval{modcoup/omega_alpha}
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but varying cutoff. Energy is normalized to the
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ergotropy. The dashed vertical lines illustrate the bath
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memory time (\(|α(τ)| = α(0)/300\)).}
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\end{subfigure}
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% \begin{subfigure}[t]{.49\linewidth}
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% \plot{modcoup/flow_interaction_overview}
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% \caption{Same situation as in \cref{fig:omega_total}. All
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% quantities are normalized to the \(ω_c=1\)
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% case.}
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% \end{subfigure}
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\begin{subfigure}[t]{.49\linewidth}
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\plot{modcoup/delta_dependence}
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\caption{Maximum one-shot power for \(10\) periods and
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different modulation frequencies \(Δ\) and coupling
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strengths.}
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\end{subfigure}
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\end{figure}
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\end{block}
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\begin{block}{Continuously Coupled Engine (Preliminary)}
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\begin{itemize}
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\item qubit coupled to two baths of different
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temperatures (\(T_c, T_h\))
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\begin{equation}
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\label{eq:antizenomodel}
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H_\sys= \frac12 \qty[ω_0 + γ Δ\sin(Δ t)]σ_z,\, L_{c,h}=\frac12 σ_x
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\end{equation}
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{\tiny \tval{anti_zeno/delta}, \tval{anti_zeno/gamma}, \tval{anti_zeno/omega_alpha},
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\tval{anti_zeno/omega_zero}, \tval{anti_zeno/tc}, \tval{anti_zeno/th}}
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\item system Hamiltonian modulated + baths periodically
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decoupled (cooldown) and ``reset''
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\end{itemize}
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\begin{figure}[H]
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\centering
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\begin{subfigure}[t]{.49\linewidth}
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\plot{anti_zeno/sd_setup}
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\caption{The spectral densities of the baths, where the vertical
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lines show where \(ω=ω_0 \pm Δ\). The overlap of the filter
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for \(n\) modulation periods is crucial for the ``anti-zeno''
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effect~\cite{Mukherjee2020Jan}.}
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\end{subfigure}
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% \begin{subfigure}[t]{.49\linewidth}
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% \plot{modcoup/flow_interaction_overview}
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% \caption{Same situation as in \cref{fig:omega_total}. All
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% quantities are normalized to the \(ω_c=1\)
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% case.}
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% \end{subfigure}
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\begin{subfigure}[t]{.49\linewidth}
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\plot{anti_zeno/modulation_setup}
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\caption{Setup of the system and coupling modulation for the
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case with ``cooldown''. The units are arbitrary. The
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initialization period and two cycles are shown.}
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\end{subfigure}
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\begin{subfigure}[t]{.49\linewidth}
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\plot{anti_zeno/anti_zeno_with_cool}
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\caption{\label{fig:cont_coup} The total energy change after the initialization
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period, where vertical lines show the points at which the
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times for the calculation of the power are taken. The mean
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power obtained is \tval{anti_zeno/power_with_cool}.}.
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\end{subfigure}
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\begin{subfigure}[t]{.49\linewidth}
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\plot{anti_zeno/anti_zeno_without_cool}
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\caption{Same as in \cref{fig:cont_coup} but without
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cooldown. The mean power obtained is
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\tval{anti_zeno/power_without_cool}. The difference is not
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significant yet.}.
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\end{subfigure}
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\end{figure}
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\end{block}
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\end{column}
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\separatorcolumn
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\end{columns}
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\end{frame}
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\end{document}
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