From d9355466302d9d1f37b37460045cf0357f8d41b7 Mon Sep 17 00:00:00 2001 From: Valentin Boettcher Date: Wed, 28 Sep 2022 17:17:39 +0200 Subject: [PATCH] some touchups on the outro --- src/outro.tex | 120 ++++++++++++++++++++++++++++---------------------- 1 file changed, 68 insertions(+), 52 deletions(-) diff --git a/src/outro.tex b/src/outro.tex index c4a6f93..c60e34c 100644 --- a/src/outro.tex +++ b/src/outro.tex @@ -3,57 +3,68 @@ In this work, we set out to find a way of accessing bath related observables, such as the expected bath energy change and the -interaction energy expectation value, using the +interaction energy expectation value using the NMQSD\footnote{Non-Markovian Quantum State Diffusion}/HOPS\footnote{Hierarchy of Pure States} framework which -we introduced in \cref{chap:intro}. This endeavor was indeed -successful as was laid out in \cref{chap:flow}. +we introduced in \cref{chap:intro}. -In \cref{chap:flow} we presented a solution to a well known model for -quantum Brownian motion. Using this solution, we were able to derive -expressions for the bath energy change \(∂_{t}\ev{H_{\bath}}\). +This endeavour was indeed successful, as we laid out in +\cref{chap:flow}. The crucial point is, that we still have access to +the bath degrees of freedom \emph{before} performing a stochastic +average over the NMQSD/HOPS trajectories. This allows us to compute +the expectation values of observables which contain the bosonic bath +operators collectively. + +In \cref{chap:flow} we presented an analytic solution to a well known +model for quantum Brownian motion. Using this solution, we were able +to derive expressions for the bath energy change +\(∂_{t}\ev{H_{\bath}}\) for models with one and two baths. This enabled us to verify the results of \cref{chap:flow} in \cref{chap:numres} by solving the same model numerically using -HOPS. Excellent agreement was found in -\cref{sec:hopsvsanalyt}. +HOPS. Excellent agreement was found in \cref{sec:hopsvsanalyt}. Turning to the spin-boson model in \cref{sec:prec_sim}, we used energy conservation to verify again, that we can consistently and efficiently -compute bath related observables with HOPS. In the cases where the -consistency condition was not met, we nevertheless found that -qualitatively correct results had been reached. The direct calculation -of the interaction energy by the use of \cref{sec:intener} gives -results that are more precise than the ones obtained through energy -conservation. +compute bath related observables with HOPS upon the example of the +interaction energy expectation value. In the cases where the +consistency condition was not met, we nevertheless found that the +results were qualitatively correct. When choosing less precise +numerical parameters, the direct calculation of the interaction energy +by the use of \cref{sec:intener} yields results that are generally +much more precise than the ones obtained through energy conservation. We continued to explore the energy transfer behavior of the zero temperature spin-boson model and found that energy transfer performance for strong coupling has a complicated dependence on the -spectral density of the bath. Energy transfer performance can be -optimized longer bath memories and resonant baths when the interaction -is turned off at the right time. +shape of the spectral density of the bath. Energy transfer performance +can be optimized through longer bath memories and resonant baths when +the interaction is switched off at the right time. Switching the +interaction off in finite time leads to cooling of the system, +especially when the steady state had been reached. -The short time dynamics of the bath energy change can be explained by -neglecting the system Hamiltonian, which we verified for the -spin-boson model. It was also found, that this short time behaviour is -already present on the trajectory level so that there are no -stochastic fluctuations for short times. During this initial period, -the auxiliary states of the HOPS are being populated. +Having explained short time dynamics of the bath energy change in +\cref{sec:pure_deph} neglecting the system Hamiltonian, which we +verified for the spin-boson model in +\cref{sec:initial-slip-sb,sec:moder-init-slip}. It was further found, +that this short time behaviour is already present on the trajectory +level so that there are no stochastic fluctuations for short +times. Instead, the auxiliary states of the HOPS are being populated +during this initial period. -In \cref{sec:singlemod} we turned to issues of quantum -thermodynamics. We reviewed some general analytical results that -bounded energy extraction from open systems in -\cref{sec:basic_thermo}, both for the single-bath and the multi-bath -case. We then turned to some more challenging applications of the HOPS -method. First, a driven spin-boson model was considered. We found that -a not insignificant fraction of the theoretical maximum of energy can -be extracted by modulating the coupling and providing a bath with long -memory time. We also demonstrated quantum friction, a quantum speed -limit and a bath resonance phenomenon. +In \cref{sec:singlemod} we turned to some issues of quantum +thermodynamics. We reviewed general analytical results that bounded +energy extraction from open systems in \cref{sec:basic_thermo}, both +for the single-bath and the multi-bath case. We then turned to some +more challenging applications of the HOPS method. First, a driven +spin-boson model was considered. We found that a not insignificant +fraction of the upper bound on the ergotropy can be extracted by +modulating the coupling and providing a bath with long memory time. We +also demonstrated quantum friction, a quantum speed limit and a bath +resonance phenomenon. -Finally, we treated a model with multiple baths in \cref{sec:otto} and -non-harmonic smooth modulation. A cyclic modulation protocol was +Finally, we treated a model with multiple baths and non-harmonic +smooth modulation in \cref{sec:otto}. A cyclic modulation protocol was implemented upon a two level system coupled to two baths in a spin-boson like fashion. We achieved finite power with finite efficiency and verified a Gibbs-like inequality @@ -66,19 +77,20 @@ optimization for optimal power which leads to the Novikov–Curzon–Ahlborn efficiency \(η_{ca}=1-\sqrt{T_{c}/T_{h}}\) is interesting in the case of stronger coupling. -Another cycle to study would be a Carnot-type cycle, where the +Another cycle to study would be a Carnot-type process, where the modulation of the system and the thermalization with the bath occur at the same time. Interpolating between Otto and Carnot, as well as studying the effect of overlapping and shifting strokes is a fascinating avenue for future exploration. -Also, more interesting working media, such as a three level system are -of interest. In \refcite{Uzdin2015Sep} it is shown, that in certain -regimes quantum coherence can lead to superior power output. In the -same regime different types heat engines are equivalent. Both these -effects have been observed experimentally in \refcite{Klatzow2019Mar}. It -would be interesting to see if the slight deviations from theory in -\cite{Klatzow2019Mar} could be explained using HOPS. +Also, more interesting working media, beginning with a three level +system, are of interest. In \refcite{Uzdin2015Sep} it is shown that +in certain regimes quantum coherence can lead to superior power +output. In the same regime different types heat engines are +equivalent. Both these effects have been observed experimentally in +\refcite{Klatzow2019Mar}. It would be interesting to see if the slight +deviations from theory in \cite{Klatzow2019Mar} could be explained +using HOPS. The so called Anti-Zeno Effect occurring in systems under fast modulation has recently received some attention @@ -86,8 +98,8 @@ modulation has recently received some attention due to the broadening of the resonance criterion which we have observed in \cref{sec:one_bath_cutoff,sec:modcoup_reso,sec:otto}. Being a -consequence of the energy time uncertainty it is being argued, that -the origin of this advantage is truly quantum. The tools for the +consequence of the energy time uncertainty it is argued that the +origin of this advantage is truly quantum. The tools for the exploitation of this effect and its verification are provided in this work. However, a strong coupling analysis has already been performed using HEOM in \refcite{Xu2022Mar}. @@ -97,13 +109,17 @@ of finite ergotropy by letting energy flow through the working medium and then extracting this ergotropy in a separate stroke. This work could be verified and expanded to the non-Markovian regime. -A useful improvement of the method would be the ability to snapshot -the total state of system and bath and then propagate this state with -different modulation protocols. Also, exploring the thermofield method -for finite temperature to avoid the slow convergence of the flow may -be worthwhile. However, at least for coupling that is not hermitian, -this would only trade computational effort for memory, as the number -of hierarchy states would increase. +A useful technical improvement of the method would be the ability to +snapshot the total state of system and bath and then propagate this +state with different modulation protocols. Also, exploring the +thermofield method for finite temperature to avoid slow convergence +would be helpful. However, at least for coupling that is not +hermitian, this would likely only trade computational effort for +memory, as the number of hierarchy states would increase. + +The splitting up of the stochastic process to calculate, for example, the +energy change of parts of the bath as discussed at the end of +\cref{sec:general_obs} is also very interesting. Finally, in the spirit of~\cite{Esposito2015Dec} one could employ the HOPS to verify whether a given definition of internal energy that