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127
references.bib
View file

@ -1663,3 +1663,130 @@
publisher = {Multidisciplinary Digital Publishing Institute},
doi = {10.3390/e24030352}
}
@article{ChengMarkovianApproximationRelaxation2005,
title = {Markovian {{Approximation}} in the {{Relaxation}} of
{{Open Quantum Systems}}},
author = {Cheng, Y. C. and Silbey, R. J.},
year = 2005,
month = nov,
journal = {The Journal of Physical Chemistry B},
volume = 109,
number = 45,
pages = {21399--21405},
issn = {1520-6106, 1520-5207},
doi = {10.1021/jp051303o},
langid = {english}
}
@article{GaspardSlippageInitialConditions1999,
ids = {Slippageinitialconditions1999},
title = {Slippage of Initial Conditions for the {{Redfield}}
Master Equation},
author = {Gaspard, P. and Nagaoka, M.},
year = 1999,
month = oct,
journal = {The Journal of Chemical Physics},
volume = 111,
number = 13,
pages = {5668--5675},
issn = {0021-9606, 1089-7690},
doi = {10.1063/1.479867},
abstract = {For a slow open quantum subsystem weakly coupled to
a fast thermal bath, we derive the general form of
the slippage to be applied to the initial conditions
of the Redfield master equation. This slippage is
given by a superoperator which describes the
non-Markovian dynamics of the subsystem during the
short-time relaxation of the thermal bath. We verify
in an example that the Redfield equation preserves
positivity after the slippage superoperator has been
applied to the initial density matrix of the
subsystem. For {$\delta$}-correlated baths, the
Redfield master equation reduces to the Lindblad
master equation and the slippage of initial
conditions vanishes consistently.},
keywords = {Matrix equations},
file = {/home/cima/Zotero/storage/9FR2GNPV/1999 - Slippage
of initial conditions for the Redfield
ma.pdf;/home/cima/Zotero/storage/VWC84D6E/Gaspard
und Nagaoka - 1999 - Slippage of initial conditions
for the Redfield
ma.pdf;/home/cima/Zotero/storage/9KVCUA3H/1.html;/home/cima/Zotero/storage/SITD7UR2/1.html}
}
@article{HaakeAdiabaticDragInitial1983,
title = {Adiabatic Drag and Initial Slip in Random Processes},
author = {Haake, Fritz and Lewenstein, Maciej},
year = 1983,
month = dec,
journal = {Physical Review A},
volume = 28,
number = 6,
pages = {3606--3612},
doi = {10.1103/PhysRevA.28.3606},
abstract = {We describe a method for the solution of
initial-value problems for random processes arising
through adiabatic approximations from Markov
processes of higher dimension. In applications to
overdamped Brownian motion and to the single-mode
laser we calculate correlation functions and discuss
initial slips.},
file = {/home/cima/Zotero/storage/LWXZ4NTQ/Haake und
Lewenstein - 1983 - Adiabatic drag and initial slip
in random
processe.pdf;/home/cima/Zotero/storage/24H57FUB/PhysRevA.28.html}
}
@article{SuarezMemoryEffectsRelaxation1992,
title = {Memory Effects in the Relaxation of Quantum Open
Systems},
author = {Su{\'a}rez, Alberto and Silbey, Robert and
Oppenheim, Irwin},
year = 1992,
month = oct,
journal = {The Journal of Chemical Physics},
volume = 97,
number = 7,
pages = {5101--5107},
issn = {0021-9606},
doi = {10.1063/1.463831},
file = {/home/cima/Zotero/storage/4QKGH7XP/Suárez et al. -
1992 - Memory effects in the relaxation of quantum
open
s.pdf;/home/cima/Zotero/storage/QNZCSDPY/1.html}
}
@article{YuPostMarkovMasterEquation2000,
title = {Post-{{Markov}} Master Equation for the Dynamics of
Open Quantum Systems},
author = {Yu, Ting and Di{\'o}si, Lajos and Gisin, Nicolas and
Strunz, Walter T.},
year = 2000,
month = feb,
journal = {Physics Letters A},
volume = 265,
number = {5-6},
pages = {331--336},
issn = 03759601,
doi = {10.1016/S0375-9601(00)00014-1},
langid = {english},
file = {/home/cima/Zotero/storage/Y9SG4FNN/Yu et al. - 2000
- Post-Markov master equation for the dynamics of
op.pdf}
}
@article{Caldeira1981Jan,
author = {Caldeira, A. O. and Leggett, A. J.},
title = {{Influence of Dissipation on Quantum Tunneling in
Macroscopic Systems}},
journal = {Phys. Rev. Lett.},
volume = 46,
number = 4,
pages = {211--214},
year = 1981,
month = jan,
issn = {1079-7114},
publisher = {American Physical Society},
doi = {10.1103/PhysRevLett.46.211}
}

221
src/num_results.tex
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@ -56,28 +56,25 @@ of this work.
\section{Some Remarks on the Methods}
\label{sec:meth}
Before we begin with the applications in earnest, let us review some
technical details.
The figures presented may feature error funnels whose origin is,
unless otherwise stated, estimated from the empirical standard
deviation of the calculated quantities due to the finite sample
size. As the quantities that are being calculated using HOPS are
essentially Monte Carlo integrals, those statistical errors scale as
\(1/\sqrt{N}\) with the sample size \(N\) and are therefore
controllable besides being simple to estimate. Note however, that a
certain number of samples is required to estimate the standard
deviation of a single trajectory correctly.
\(1/\sqrt{N}\) with the sample size \(N\) and are controllable besides
being simple to estimate. Note however, that a certain number of
samples is required to estimate the standard deviation of a single
trajectory correctly.
To tell whether some vector quantities\footnote{For example a time
series.} \(X_1, X_2\) obtained with HOPS or otherwise are compatible
with each other or an analytical result, we consider the quantity
\(Δ=X_1 - X_2\). Assuming all numerical errors are negligible, we
demand that \(\abs{Δ} \leq σ_Δ\) for at least
\(68\%\)\footnote{Roughly one standard deviation of a Gaussian random
variable.} of the entries of the \(X_i\), where \(σ_Δ\) is the
standard deviation due to the Monte Carlo sampling. This percentage is
often displayed in legends as a number in parentheses.
series.} \(X_1, X_2\) obtained numerically or otherwise are
compatible with each other we introduce the quantity \(Δ=X_1 -
X_2\). Assuming all numerical errors are negligible, we demand that
\(\abs{Δ} \leq σ_Δ\) for at least \(68\%\)\footnote{Roughly one
standard deviation of a Gaussian random variable.} of the entries
of the \(X_i\), where \(σ_Δ\) is the standard deviation due to the
Monte Carlo sampling. This percentage is displayed as a number in
parentheses in legends of plots in the following sections.
For the estimation of mean and standard deviation from trajectory
data, Welford's online algorithm is employed to avoid catastrophic
@ -88,21 +85,19 @@ In all simulations discussed use an Ohmic spectral density
\label{eq:ohmic_sd}
J(ω)=η ω \eu^{-\frac{ω}{ω_{c}}}\quad (ω>0)
\end{equation}
is used unless otherwise. This spectral density models an environment
with a physical energy spectrum that is bounded from below and allows
the application of the finite temperature method described
in~\cite{RichardDiss} and \cref{sec:lin_finite}. Also, \(J(0) = 0\)
ensures that there is a unique zero temperature state of the
bath. In~\cite{Kolar2012Aug} it is argued (under weak coupling
assumptions), that \(J(ω)\approx ω^γ\) with \(γ<1\) could lead to a
violation of the third law. Physically, a scaling of the spectral
density \(\propto ω\) is connected to acoustic
phonons~\cite{Kolar2012Aug}.
is used. In \cref{eq:ohmic_sd} \(η\) is a scaling constant and
\(ω_c\) (the cutoff frequency) regulates the decay of the spectral
density. Due to ambiguities that can arise when using different
normalization schemes for the BCF as was the case for this work, we
express the coupling strength in terms of \(α(0)\) rather than giving
values for \(η\). In the convention used above, the parameter \(η\)
may be recovered as follows
\begin{equation}
\label{eq:get_eta_back}
η=\frac{α(0) π}{ω_{c}^{2}}.
\end{equation}
In \cref{eq:ohmic_sd} \(η\) is a scaling
constant and \(ω_c\) (the cutoff frequency) regulates the decay of the
spectral density. The corresponding bath correlation function (BCF)
is
The corresponding bath correlation function (BCF) is
\begin{equation}
\label{eq:ohmic_bcf}
α(τ) = \frac{1}{π} ∫\dd{ω} J(ω) \eu^{-\iu ωτ} =
@ -112,6 +107,17 @@ We see that higher cutoff frequencies correspond to a faster decay of
the bath correlation function. This parameter provides control over
the ``Markovianity'' of the bath.
The ohmic spectral density models an environment with a physical
energy spectrum that is bounded from below and allows the application
of the finite temperature method described in~\cite{RichardDiss} and
\cref{sec:lin_finite}. Also, \(J(0) = 0\) ensures that there is a
unique zero temperature state of the bath. In~\cite{Kolar2012Aug} it
is argued (under weak coupling assumptions), that \(J(ω)\approx ω^γ\)
with \(γ<1\) could lead to a violation of the third law. Physically,
a scaling of the spectral density \(\propto ω\) is connected to
acoustic phonons~\cite{Kolar2012Aug}.
It may be remarked, that~\cref{eq:ohmic_bcf} does not correspond to a
simple sum of exponentials. As such it exercises the HOPS method and
serves as a model for a general bath correlation function. For use
@ -138,24 +144,26 @@ bath energy flow.
We begin with the simplest possible model of a harmonic oscillator
coupled to a single zero temperature bath. For the simulations with
HOPS the model \cref{eq:one_ho_hamiltonian} was made dimensionless by
choosing \(Ω=1\). Simulations were run for both for zero temperature
and a finite temperature with varying bath correlation functions.
choosing \(Ω=1\). Simulations were run for both zero temperature and
finite temperature with various parameters of the bath correlation
function.
\begin{figure}[t]
\centering
\includegraphics{figs/analytic_comp/flow_comp_zero.pdf}
\caption{\label{fig:comp_zero_t} The bath energy flow \(-J\) for
different parameters of the ohmic bath correlation function
\cref{eq:ohmic_bcf}. The right panel shows the short time behaviour
of the flow. The solid lines have been obtained with HOPS and the
dashed lines using the analytic solution. A good agreement is
evident visually and corroborated by the consistency values in the
legend (see \cref{sec:meth} for an explanation). The simulation
with \(ω_{c}=3\) (orange) stands out for its slow long term decay,
\cref{eq:ohmic_bcf}. The right panel shows the short time
behaviour of the flow. The solid lines have been obtained with
HOPS and the dashed lines using the analytic solution. A
strikingly good agreement is evident visually and corroborated by
the consistency values in parentheses the legend (see
\cref{sec:meth} for an explanation). The simulation with
\(ω_{c}=3\) (orange) stands out for its slow long term decay,
whereas the same simulation with longer bath correlation time
(blue) initially decays much slower but falls below its orange
(blue) decays much slower initially but falls below its orange
counterpart for longer times.}
\end{figure}
\paragraph{Zero Temperature}
The bath energy flow \(J=-∂_t\ev{H_\bath}\), from here on called
simply ``the flow'' or ``bath energy flow'', for the zero temperature
@ -173,13 +181,13 @@ The simulation was run with a hierarchy depth of
harmonic oscillator Hilbert space was truncated to \(15\) dimensions
in the energy basis. As the initial state the first excited state of
the oscillator was chosen so that some nontrivial energy flow would
occur. Some \(N=5000\) trajectories have been computed and lead to a
quite satisfactory statistical error that is small enough to be
invisible in \cref{fig:comp_zero_t}. The normalized standard deviation
of the bath energy flow follows the usual one-over-square-root rule as
is illustrated in \cref{fig:sqrt_conv}. Even after just \(N=1000\)
trajectories the normalized statistical error is on the order of
\(10^{-3}\).
occur. A reasonable number of \(N=5000\) trajectories have been
computed and lead to a quite satisfactory statistical error that is
small enough to be invisible in \cref{fig:comp_zero_t}. The normalized
standard deviation of the bath energy flow follows the usual
one-over-square-root rule as is illustrated in
\cref{fig:sqrt_conv}. Even after just \(N=1000\) trajectories the
normalized statistical error is on the order of \(10^{-3}\).
\begin{figure}[htp]
\centering
\includegraphics{figs/analytic_comp/sqrt_convergence.pdf}
@ -218,18 +226,20 @@ separate fit was made rather than using the fit from
Interestingly, the solutions using a BCF expansion with three terms or
fewer lead to an unphysical non-zero steady state bath energy
flow. Considering specifically the case of one expansion term, this
may be related to the fact that now the BCF term so that
\(α(τ)=G \exp(-Wτ)\) is related to a Lorentzian spectral density that
also includes unphysical negative frequencies. To correctly reproduce
the steady state, the fit must model the decay of the BCF on the
appropriate time scales.
may be related to the fact that now the corresponding spectral density
of \(α(τ)=G \exp(-Wτ)\) also has a finite value for negative
frequencies and will have negative parts when \(\Im G \neq 0\), making
it unphysical. In general, the fit must model the decay of the BCF on
the appropriate time scales, to correctly reproduce the steady state.
Although the simulations are primarily intended as a benchmark for
HOPS and a verification for the results of \cref{chap:flow} some
observations can be made in \cref{fig:comp_zero_t}.
First, the flows for different parameters all feature the
characteristic spike in the flow, not unlike an ``initial slip''. As
characteristic spike in the flow, not unlike the phenomenon known and
discussed in the literature as ``initial
slip''~\cite{ChengMarkovianApproximationRelaxation2005,GaspardSlippageInitialConditions1999,HaakeAdiabaticDragInitial1983,SuarezMemoryEffectsRelaxation1992,YuPostMarkovMasterEquation2000}. As
the initial state is a product state which is unaware of the
bath-system interaction, the state undergoes some short time dynamics
to compensate the change of the effective oscillator dynamics due to
@ -255,10 +265,10 @@ takes the form \cite{Weiss2008Mar}
H_{\mathrm{C}} \sim ∑_{λ}\frac{\abs{g_{λ}}^{2}}{ω_{λ}} q^{2}.
\end{equation}
When \cref{eq:counter_term} is included in \cref{sec:oneosc}, our
model becomes the Caldiera-Legget Model. The counter term balances
the renormalization of the system potential due to the bath. It is
therefore an interesting question how the initial slip behaves in its
presence.
model becomes the Caldiera-Legget Model~\cite{Caldeira1981Jan}. In
many cases, the counter term balances the renormalization of the
system potential due to the bath. It is therefore an interesting
question how the initial slip behaves in its presence.
The time dependence of the flow also varies both with the shape of the
BCF and the coupling strength. For longer correlation times
@ -270,9 +280,9 @@ situation is reversed and the simulation cuttoff frequency \(ω_{c}=3\)
strengths we can observe a ``backflow'' of energy out of the bath as
can be observed for the red line in the right panel of
\cref{fig:comp_zero_t}. In all cases the flow features some
oscillations and decays to zero which is physical for the situation of
a harmonic oscillator that loses most of its energy into a zero
temperature bath.
oscillations and decays to zero which is physically reasonable for the
situation of a harmonic oscillator that loses most of its energy into
a zero temperature bath.
When the bath memory is short however, the short-term energy flow is
of less magnitude. Correspondingly, the system energy decays slower
@ -293,17 +303,18 @@ in frequency space.
dynamics. The strongest coupling simulation (red) converges to a
markedly different steady state.}
\end{figure}
The ``backflow'' observed in \cref{fig:comp_zero_t} for stronger
coupling seems to diminish the advantage of stronger coupling over
larger bath memories. The decay of the red curve in the right panel of
\cref{fig:ho_zero_entropy} is barely faster than the decay of the blue
curve, despite much larger coupling strength in the red case. This may
be due to the system interacting ``too long'' with a given portion of
the bath. Here we observe this behaviour for stronger coupling which
shortens the timescale of the energy exchange. In
\cref{sec:one_bath_cutoff} we will see, that these oscillations occur
generically for long bath memories and stronger coupling and how they
might be exploited.
The ``backflow'' of energy out of the bath observed in
\cref{fig:comp_zero_t} for stronger coupling seems to diminish the
advantage of stronger coupling over larger bath memories when it comes
to reducing the system energy. The decay of the system energy of the
model with the largest coupling strength \(α(0)\) (red curve) in the
right panel of \cref{fig:ho_zero_entropy} is barely faster than the
decay of the energy of the more weakly coupled model with longer
memory (blue curve), despite much larger coupling. Here backflow is
observed for stronger coupling which shortens the timescale of the
energy exchange. In \cref{sec:one_bath_cutoff} we will see, that this
phenomenon generically occurs for long bath memories in combination
with stronger coupling and learn how they might be exploited.
Note also, that the steady state is not a product state as can be
determined from the residual entropy in \cref{fig:ho_zero_entropy} for
@ -317,7 +328,7 @@ qualitatively different steady state than the one with the same cutoff
but weaker coupling strength (green line). This also manifests in a
higher expected system energy in the steady state.
\begin{wrapfigure}[18]{o}{0.4\textwidth}
\begin{wrapfigure}[-4]{O}{0.4\textwidth}*
\centering
\includegraphics{figs/analytic_comp/timescale_comparison}
\caption{\label{fig:timescale_comp} A comparison of bath vs
@ -326,18 +337,19 @@ higher expected system energy in the steady state.
in \cref{fig:ho_zero_entropy}. The orange line is set far apart
from the other simulations.}
\end{wrapfigure}
The time dependence of the entropy the expectation value of the system
energy is markedly different for the cutoff \(ω_c=3\) (orange
line). Although the coupling strength is similar to the
The time dependence of the system entropy and the system energy
expectation value is markedly different for the cutoff \(ω_c=3\)
(orange line). Although the coupling strength is similar to the
\(α(0)=0.32,\, ω_c=1\) case (blue line) the energy loss of the system
is much slower and the initial energy gain is less pronounced. This is
consistent with the flow in \cref{fig:comp_zero_t}. The case is set
apart by the long interaction timescale \(τ_{\inter}\) due to the
weaker coupling strength \(\sim α(0)^{-1/2}\) and the short bath
timescale \(τ_{\bath}\sim ω_{c}^{-1}\). Of course absolute values of
these time scales of no great value here. A comparison of relative
time scales \(\sqrt{α(0)}/ω_{c}\sim τ_{\bath}/τ_{\inter}\) can be
found in \cref{fig:timescale_comp} supporting the above argument.
consistent with the flow in \cref{fig:comp_zero_t} which has a smaller
overall magnitude. The case is set apart by the long interaction
timescale \(τ_{\inter}\) due to the weaker coupling strength
\(\sim α(0)^{-1/2}\) and the short bath timescale
\(τ_{\bath}\sim ω_{c}^{-1}\). Of course absolute values of these time
scales are of no great value here. A comparison of relative time
scales \(\sqrt{α(0)}/ω_{c}\sim τ_{\bath}/τ_{\inter}\) can be found in
\cref{fig:timescale_comp} supporting the above argument.
\paragraph{Finite Temperature}
\begin{figure}[t]
@ -445,28 +457,28 @@ and compare the results with the analytical solution.
For simplicity, the parameters were chosen symmetric so that the
frequencies of both oscillators are the same \(Ω=Λ=1\). As before,
\(Ω\) defines the energy unit. The zero temperature bath correlation
functions of both baths were chosen identically with a cutoff
frequency \(ω_c=2\). The intra-oscillator coupling was chosen as
\(γ=0.5\). The hierarchy was truncated so that \(\abs{\vb{k}}\leq 3\)
and a BCF expansion with five terms was chosen to limit memory demands
and \(10^{4}\) trajectories were integrated.
functions of both baths were identical with a cutoff frequency
\(ω_c=2\) and the intra-oscillator coupling was set to \(γ=0.5\). The
hierarchy was truncated so that \(\abs{\vb{k}}\leq 3\) and a BCF
expansion with five terms was employed to limit memory demands. For
adequate convergence \(10^{4}\) trajectories were integrated.
To limit the variance the temperature of one of the baths was set to
zero, so that only one thermal stochastic process was introduced. The
other bath was chosen to have \(T=0.6\). The ground state of the
system Hamiltonian \(\ket{0}\otimes \ket{0}\) was chosen as the
initial state of the oscillators.
To limit the variance, the temperature of one of the baths was set to
zero, so that only one thermal stochastic process was introduced. The
other bath was initialized with the temperature \(T=0.6\) and the
initial state of the oscillators was chosen to be the ground state of
the system Hamiltonian \(\ket{0}\otimes \ket{0}\).
The main challenge of simulating the model
\cref{eq:hamiltonian_two_bath} is the dimension of the system Hilbert
space which is constrained by the available memory. In the simulation
discussed here, each oscillator was truncated at \(9\) levels leading
to \(9^2 = 81\) dimensions in total\footnote{This is a naive method of
truncation, but sufficient for the purposes of this work.}. The
effect of a too drastic truncation of the system Hilbert space can be
seen in \cref{fig:insufficient_levels}. At the temperature chosen, the
mean level occupation of a harmonic oscillator is given by the Bose
distribution
to a Hilbert space dimension of \(9^2 = 81\) in total\footnote{This is
a naive method of truncation, but sufficient for the purposes of
this work.}. The effect of a too drastic truncation of the system
Hilbert space can be seen in \cref{fig:insufficient_levels}. At the
temperature chosen, the mean level occupation of a harmonic oscillator
is given by the Bose distribution
\begin{equation}
\label{eq:harm_mean_occ}
\ev{n} = \frac{1}{\eu^{Ωβ}-1} \approx 0.23 < 1.
@ -484,12 +496,16 @@ sufficient to corroborate the validity of the results of
\cref{sec:multibath}. With more computational effort\footnote{Mainly
more BCF expansion terms.} and fine-tuning of parameters an even
better agreement between the analytical and the numerical results may
be achieved.
be achieved. Note that the consistency figure given for the hot bath
is higher, as the statistical error bounds are greater due to the
finite temperature. These increased error bounds allow for more
deviation.
\begin{figure}[htp]
\centering
\begin{subfigure}[t]{.49\linewidth}
\includegraphics{figs/analytic_comp/comparison_two_5bcf_5ho.pdf}
\caption{\label{fig:insufficient_levels}\(\dim\hilb_\sys=25\).}
\caption{\label{fig:insufficient_levels}\(\dim\hilb_\sys=25\). No
agreement between the analytical solution and HOPS.}
\end{subfigure}
\begin{subfigure}[t]{.49\linewidth}
\includegraphics{figs/analytic_comp/comparison_two_ho.pdf}
@ -528,12 +544,15 @@ deviation of a given observable. In \cref{sec:pure_deph} we will
discuss the short term dynamics of our general model and explain the
peak in the flow, as seen in \cref{fig:comp_zero_t}.
For future work there remains the generalization of the work in this
section and \cref{chap:analytsol} to time dependent couplings and
For future work there remains the generalization of this section and
\cref{chap:analytsol} to time dependent couplings and
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.
Let us now turn to the short time behavior of the flow and the initial
slip in \cref{sec:pure_deph} .
\section{Pure Dephasing and the Initial Slip}
\label{sec:pure_deph}
As seen in \cref{fig:comp_finite_t,fig:comp_zero_t,fig:comp_two_bath},