mirror of
https://github.com/vale981/master-thesis
synced 2025-03-04 17:41:43 -05:00
tex initial slip theory
This commit is contained in:
parent
78805b86b2
commit
ccc590d996
2 changed files with 258 additions and 0 deletions
|
|
@ -419,6 +419,88 @@ This essentially boils down to the replacement
|
|||
where the quantities involved are as in \fixme{reference} and
|
||||
\cref{eq:xiproc}.
|
||||
|
||||
\section{Pure Dephasing: The initial Slip}
|
||||
\label{sec:pure_deph}
|
||||
|
||||
As seen in \fixme{include plots}, the short time behavior of the bath
|
||||
energy flow is dominated by characteristic peak at short
|
||||
times. Because this peak occurs at very short time scales, it may in
|
||||
part be explained by a simple calculation which neglects the system
|
||||
dynamics, setting \(H_\sys=0\).
|
||||
|
||||
We solve the model with the Hamiltonian (Schr\"odinger picture)
|
||||
\begin{equation}
|
||||
\label{eq:puredeph}
|
||||
H = L^†(t) B + L(t) B^† + H_\bath
|
||||
\end{equation}
|
||||
with \(L(t)=L(t)^†\), \([L(t), L(s)] = 0\;\forall t,s\) (so that
|
||||
Heisenberg Hamiltonian matches \cref{eq:puredeph}) and \(B,H_\bath\)
|
||||
as in \cref{eq:bop}.
|
||||
|
||||
Because \([L,H]=0\) we can immediately solve \(L_H(t)=L_S(t)\), where
|
||||
the subscript signify the Heisenberg and Schr\"odinger pictures
|
||||
respectively. The Heisenberg equations for the \(a_λ\) yield
|
||||
\begin{equation}
|
||||
\label{eq:alapuredeph}
|
||||
a_λ(t) = a_λ(0) \eu^{-\iu ω_λ t} - \iu g_λ^\ast∫_0^t\dd{s} L(s)
|
||||
\eu^{-\iu ω_λ (t-s)}.
|
||||
\end{equation}
|
||||
|
||||
This allows us to calculate
|
||||
\begin{equation}
|
||||
\label{eq:pureflow}
|
||||
\dot{H}_\bath = - ∑_λ g_λ L(t) \qty[∂_t a_λ(0) \eu^{\iu ω_λ t} - \iu
|
||||
g_λ^\ast∫_0^t\dd{s} L(s) ∂_t \eu^{-\iu ω_λ (t-s)}] + \hc,
|
||||
\end{equation}
|
||||
which gives with a state of the form \(ρ=\ketbra{ψ} \otimes ρ_β\)
|
||||
(\(ρ_β\) being a thermal state)
|
||||
\begin{equation}
|
||||
\label{eq:pureflowexpectation}
|
||||
\ev{\dot{H}_\bath } = -2 ∫_0^t\dd{s}\ev{L(t)L(s)} \Im[\dot{α}(t-s)].
|
||||
\end{equation}
|
||||
|
||||
For time independent \(L\) this becomes
|
||||
\begin{equation}
|
||||
\label{eq:pureflowtimeindep}
|
||||
\ev{\dot{H}_\bath } = 2 \ev{L^2} \Im[\dot{α}(t)].
|
||||
\end{equation}
|
||||
|
||||
The proportionality to the imaginary BCF \(α\) does explain the
|
||||
initial peak in the bath energy flow. The imaginary part of the BCF is
|
||||
zero for \(t=0\) and then usually features a peak at rather short
|
||||
times (assuming finite correlation times). For the ohmic BCF used
|
||||
here, this feature is very prominent.
|
||||
\fixme{insert graph}
|
||||
|
||||
Interestingly, \cref{eq:pureflowexpectation} does not contain any
|
||||
reference to the temperature of the bath. Therefore, the bath energy
|
||||
can only surpass its initial value in this model, as the dynamics
|
||||
match the zero temperature case in which the bath has minimal energy
|
||||
in the initial state. A thermodynamically useful model should
|
||||
therefore feature an significant system dynamics that do not commute
|
||||
with the interaction or fast modulation so that the Hamiltonian does
|
||||
not commute with itself at different times. The latter may induce
|
||||
deviations from the pure-dephasing behavior at very short time scales
|
||||
and thus be useful for finite power output. \fixme{here the plot with
|
||||
energy extraction would be good.} Coupling that is not self-adjoint
|
||||
\fixme{plot} may also have this effect, but may be harder to
|
||||
physically motivate. For the spin-boson system it is the result of the
|
||||
random wave approximation, which however does not imply weak
|
||||
coupling~\cite{Irish2007Oct}.
|
||||
|
||||
For completeness, the interaction energy is given by
|
||||
\begin{equation}
|
||||
\label{eq:pureinter}
|
||||
H_\inter = L(t)\qty[∑_λg_λ\qty(a_λ(0)\eu^{-\i ω_λ t} - \i
|
||||
g^\ast_λ∫_0^t\dd{s} L(s) \eu^{\i ω_λ (t-s)})] + \hc,
|
||||
\end{equation}
|
||||
yielding
|
||||
\begin{equation}
|
||||
\label{eq:pureinter}
|
||||
\ev{H_\inter} = 2 ∫_0^t\dd{s}\ev{L(t)L(s)} \Im[α(t-s)].
|
||||
\end{equation}
|
||||
\fixme{plots}
|
||||
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
|
|
|
|||
|
|
@ -743,3 +743,179 @@ url = { http://slubdd.de/katalog?TN_libero_mab2147995 }
|
|||
eprint = {2009.00485},
|
||||
doi = {10.1103/PhysRevApplied.15.064074}
|
||||
}
|
||||
|
||||
@article{Talkner2020Oct,
|
||||
author = {Talkner, Peter and H{\ifmmode\ddot{a}\else\"{a}\fi}nggi, Peter},
|
||||
title = {{Colloquium: Statistical mechanics and thermodynamics at strong coupling: Quantum and classical}},
|
||||
journal = {Rev. Mod. Phys.},
|
||||
volume = {92},
|
||||
number = {4},
|
||||
pages = {041002},
|
||||
year = {2020},
|
||||
month = {Oct},
|
||||
issn = {1539-0756},
|
||||
publisher = {American Physical Society},
|
||||
doi = {10.1103/RevModPhys.92.041002}
|
||||
}
|
||||
@article{Lenard1978Dec,
|
||||
author = {Lenard, A.},
|
||||
title = {{Thermodynamical proof of the Gibbs formula for elementary quantum systems}},
|
||||
file = {~/bibliography/bibtex-pdfs/Lenard1978Dec.pdf},
|
||||
journal = {J. Stat. Phys.},
|
||||
volume = {19},
|
||||
number = {6},
|
||||
pages = {575--586},
|
||||
year = {1978},
|
||||
month = {Dec},
|
||||
issn = {1572-9613},
|
||||
publisher = {Kluwer Academic Publishers-Plenum Publishers},
|
||||
doi = {10.1007/BF01011769}
|
||||
}
|
||||
|
||||
@book{Binder2018,
|
||||
author = {Binder, Felix and Correa, Luis A. and Gogolin, Christian and Anders, Janet and Adesso, Gerardo},
|
||||
title = {{Thermodynamics in the Quantum Regime}},
|
||||
year = {2018},
|
||||
issn = {0168-1222},
|
||||
isbn = {978-3-319-99045-3},
|
||||
publisher = {Springer},
|
||||
address = {Cham, Switzerland},
|
||||
doi = {10.1007/978-3-319-99046-0}
|
||||
}
|
||||
@article{Mu2017Dec,
|
||||
author = {Mu, Anqi and Agarwalla, Bijay Kumar and Schaller, Gernot and Segal, Dvira},
|
||||
title = {{Qubit absorption refrigerator at strong coupling}},
|
||||
file = {~/bibliography/bibtex-pdfs/Mu2017Dec.pdf},
|
||||
journal = {New J. Phys.},
|
||||
volume = {19},
|
||||
number = {12},
|
||||
pages = {123034},
|
||||
year = {2017},
|
||||
month = dec,
|
||||
issn = {1367-2630},
|
||||
publisher = {IOP Publishing},
|
||||
doi = {10.1088/1367-2630/aa9b75}
|
||||
}
|
||||
@article{Wiedmann2021Jun,
|
||||
author = {Wiedmann, Michael and Stockburger, J{\ifmmode\ddot{u}\else\"{u}\fi}rgen T. and Ankerhold, Joachim},
|
||||
title = {{Non-Markovian quantum Otto refrigerator}},
|
||||
file = {~/bibliography/bibtex-pdfs/Wiedmann2021Jun.pdf},
|
||||
journal = {Eur. Phys. J. Spec. Top.},
|
||||
volume = {230},
|
||||
number = {4},
|
||||
pages = {851--857},
|
||||
year = {2021},
|
||||
month = jun,
|
||||
issn = {1951-6401},
|
||||
publisher = {Springer Berlin Heidelberg},
|
||||
doi = {10.1140/epjs/s11734-021-00094-0}
|
||||
}
|
||||
@article{Geva1992Feb,
|
||||
author = {Geva, Eitan and Kosloff, Ronnie},
|
||||
title = {{A quantum{-}mechanical heat engine operating in finite time. A model consisting of spin{-}1/2 systems as the working fluid}},
|
||||
file = {~/bibliography/bibtex-pdfs/Geva1992Feb.pdf},
|
||||
journal = {J. Chem. Phys.},
|
||||
volume = {96},
|
||||
number = {4},
|
||||
pages = {3054--3067},
|
||||
year = {1992},
|
||||
month = feb,
|
||||
issn = {0021-9606},
|
||||
publisher = {American Institute of Physics},
|
||||
doi = {10.1063/1.461951}
|
||||
}
|
||||
@article{Feldmann2004Oct,
|
||||
author = {Feldmann, Tova and Kosloff, Ronnie},
|
||||
title = {{Characteristics of the limit cycle of a reciprocating quantum heat engine}},
|
||||
file = {~/bibliography/bibtex-pdfs/Feldmann2004Oct.pdf},
|
||||
journal = {Phys. Rev. E},
|
||||
volume = {70},
|
||||
number = {4},
|
||||
pages = {046110},
|
||||
year = {2004},
|
||||
month = oct,
|
||||
issn = {2470-0053},
|
||||
publisher = {American Physical Society},
|
||||
doi = {10.1103/PhysRevE.70.046110}
|
||||
}
|
||||
@article{Raja2021Mar,
|
||||
author = {Raja, S. Hamedani and Maniscalco, S. and Paraoanu, G. S. and Pekola, J. P. and Gullo, N. Lo},
|
||||
title = {{Finite-time quantum Stirling heat engine}},
|
||||
file = {~/bibliography/bibtex-pdfs/Raja2021Mar.pdf},
|
||||
journal = {New J. Phys.},
|
||||
volume = {23},
|
||||
number = {3},
|
||||
pages = {033034},
|
||||
year = {2021},
|
||||
month = mar,
|
||||
issn = {1367-2630},
|
||||
publisher = {IOP Publishing},
|
||||
doi = {10.1088/1367-2630/abe9d7}
|
||||
}
|
||||
@article{Kofman2004Sep,
|
||||
author = {Kofman, A. G. and Kurizki, G.},
|
||||
title = {{Unified Theory of Dynamically Suppressed Qubit Decoherence in Thermal Baths}},
|
||||
file = {~/bibliography/bibtex-pdfs/Kofman2004Sep.pdf},
|
||||
journal = {Phys. Rev. Lett.},
|
||||
volume = {93},
|
||||
number = {13},
|
||||
pages = {130406},
|
||||
year = {2004},
|
||||
month = sep,
|
||||
issn = {1079-7114},
|
||||
publisher = {American Physical Society},
|
||||
doi = {10.1103/PhysRevLett.93.130406}
|
||||
}
|
||||
@article{Scopa2018Jun,
|
||||
author = {Scopa, Stefano and Landi, Gabriel T. and Karevski, Dragi},
|
||||
title = {{Lindblad-Floquet description of finite-time quantum heat engines}},
|
||||
file = {~/bibliography/bibtex-pdfs/Scopa2018Jun.pdf},
|
||||
journal = {Phys. Rev. A},
|
||||
volume = {97},
|
||||
number = {6},
|
||||
pages = {062121},
|
||||
year = {2018},
|
||||
month = jun,
|
||||
issn = {2469-9934},
|
||||
publisher = {American Physical Society},
|
||||
doi = {10.1103/PhysRevA.97.062121}
|
||||
}
|
||||
@book{Kurizki2021Dec,
|
||||
author = {Kurizki, Gershon and Kofman, Abraham G.},
|
||||
title = {{Thermodynamics and Control of Open Quantum Systems}},
|
||||
journal = {Cambridge Core},
|
||||
year = {2021},
|
||||
month = dec,
|
||||
isbn = {978-1-31679845-4},
|
||||
publisher = {Cambridge University Press},
|
||||
address = {Cambridge, England, UK},
|
||||
doi = {10.1017/9781316798454}
|
||||
}
|
||||
|
||||
@article{Uzdin2015Sep,
|
||||
author = {Uzdin, Raam and Levy, Amikam and Kosloff, Ronnie},
|
||||
title = {{Equivalence of Quantum Heat Machines, and Quantum-Thermodynamic Signatures}},
|
||||
journal = {Phys. Rev. X},
|
||||
volume = {5},
|
||||
number = {3},
|
||||
pages = {031044},
|
||||
year = {2015},
|
||||
month = sep,
|
||||
issn = {2160-3308},
|
||||
publisher = {American Physical Society},
|
||||
doi = {10.1103/PhysRevX.5.031044}
|
||||
}
|
||||
|
||||
@article{Irish2007Oct,
|
||||
author = {Irish, E. K.},
|
||||
title = {{Generalized Rotating-Wave Approximation for Arbitrarily Large Coupling}},
|
||||
journal = {Phys. Rev. Lett.},
|
||||
volume = {99},
|
||||
number = {17},
|
||||
pages = {173601},
|
||||
year = {2007},
|
||||
month = oct,
|
||||
issn = {1079-7114},
|
||||
publisher = {American Physical Society},
|
||||
doi = {10.1103/PhysRevLett.99.173601}
|
||||
}
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue