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7.3 KiB
7.3 KiB
- Literature
- Important Basics
- Todo
- Where is stochastic unraveling explained in more detail?
- RESOLVED Ito formalism necessary?
- RESOLVED the stochastic calculus…
- understanding NMQSD
- How are gaussian processes described by their autocorellation
- Which mean is meant in the Bath Correlation Function?
- What is the justification for substitutiong zt for a stochastic process?
- Why in the first place? -> sampling -> but why processes
- The langevin eq. for Q in cite:Strunz2001Habil is NO LANGEVIN equation?!
- Mathematical nitpicks in cite:Strunz2001Habil
- IN cite:Strunz2001Habil this is meant as integral over initial conditions?
- ASK Quantum Fluctuation theorems?
- Submit stocproc and … patches
- ASK Only β dependence in Rivas H from definition, or also through time development?
- ASK Nonlinear woes!
- Tasks
Literature
Stochastic Processes
Open Systems
- Open Quantum Systems by Rivas
- Fundamentals of quantum optics benjamin by Klauder
Stochastic Unravelings
- The quantum-state diffusion model applied to open systems one of the first applications
- Decoherent histories and quantum state diffusion
NMQSD
HOPS
Numerik
Important Basics
Todo
DONE Where is stochastic unraveling explained in more detail?
-
maybe in sources 1-7 in the cite:Diosi1997
- cite:Diosi1995Jan
- wait for hab…
RESOLVED Ito formalism necessary?
RESOLVED the stochastic calculus…
DONE understanding NMQSD
DONE How are gaussian processes described by their autocorellation
DONE Which mean is meant in the Bath Correlation Function?
- ok mean in initial state
DONE What is the justification for substitutiong zt for a stochastic process?
- actually we do not really substitute -> the sample trajectories are a stoch. process
DONE Why in the first place? -> sampling -> but why processes
DONE The langevin eq. for Q in cite:Strunz2001Habil is NO LANGEVIN equation?!
- well sort of. the solution is correct
DONE Mathematical nitpicks in cite:Strunz2001Habil
DONE IN cite:Strunz2001Habil this is meant as integral over initial conditions?
DONE Submit stocproc and … patches
ASK Only β dependence in Rivas H from definition, or also through time development?
ASK Nonlinear woes!
- derivative of D operator?
- Heisenberg Method can't work. At least it's no linear operator
- ahh see the end of my notes
Tasks
DONE Implement Basic HOPS
CLOCK: [2021-10-08 Fri 08:51] CLOCK: [2021-10-07 Thu 13:38]–[2021-10-07 Thu 17:50] => 4:12
- see my stoch. proc experiments
- ill use richards package
Find the Steady State
Quantify Heat Transfer
- not as easy as in the cite:Kato2015Aug paper
- maybe heisenberg picture useful
- see my notes. just calculate the time derivative of the bath energy expectation
- my first experiments yield bogus numerics…
- richards code makes it work
-
for derivations see
- the energy balance checks out System + Interaction Energy and my notes
- i've generalized to multiple exponential in this document
DONE TeX notes
- done with nonlinear
TODO verify that second hops state vanishes
DONE Try to get Richards old HOPS working
- code downloaded from here
- it works see Energy Flow
- interestingly with this model: only one aux state
DONE Test Nonlinear hops
- see here
TODO Generalize to two Baths
- bath-bath correlations
TODO Generalize to Nonzero Temp
- in cite:RichardDiss the noise hamiltonian method is described
- b.c. only on system -> calculation should go through :)
- not that easy, see notes
- includes time derivative of stoch proc
- idea: sample time derivative and integrate
- not as bad as thought: no exponential form needed -> process smooth
- one can get around the time derivative
- i have implemented finite temperature here
TODO Analytic Verification
- cummings
- and pseudo-mode
DONE figure out why means involving the stoch. process are so bad
- maybe y is wrong -> no
- then: not differentiable + too noisy
- other term is integral and continous, converges faster?
- my test with the gauss process was tupid -> no sum of exponentials
- it works with proper smooth process: Energy Flow in the linear case with smooth correlation…
ASK
- why do i have to take the conjugate of the process?
DONE VORTRAG
- https://www.youtube.com/watch?v=5bRii85RT8s&list=PLJfdTiUFX4cNiK44-ScthZC2SNNrtUGu1&index=33;
- where do i find out more about \(C^\ast\) algebras?
- power \(\dot{W}(t):=\frac{d}{d t}\langle H(t)\rangle=\operatorname{Tr}\left[\dot{H}_{\mathrm{S}}(t) \rho_{\mathrm{SR}}(t)\right]=\operatorname{Tr}\left[\dot{H}_{\mathrm{S}}(t) \rho_{\mathrm{S}}(t)\right]\)
- work is just the change of total energy
- Definitions \(H_{\mathrm{S}}^{\circledast}(t, \beta):=-\beta^{-1} \log \left[\Lambda_{t} \mathrm{e}^{-\beta H_{\mathrm{S}}}\right]\left\{\begin{array}{l}E_{\mathrm{int}}(t):=\operatorname{Tr}\left\{\rho_{\mathrm{S}}(t)\left[H_{\mathrm{S}}^{\circledast}(t, \beta)+\beta \partial_{\beta} H_{\mathrm{S}}^{\circledast}(t, \beta)\right]\right\} \\ F(t):=\operatorname{Tr}\left\{\rho_{\mathrm{S}}(t)\left[H_{\mathrm{S}}^{\circledast}(t, \beta)+\beta^{-1} \log \rho_{\mathrm{S}}(t)\right]\right\} \\ S(t):=\operatorname{Tr}\left\{\rho_{\mathrm{S}}(t)\left[-\log \rho_{\mathrm{S}}(t)+\beta^{2} \partial_{\beta} H_{\mathrm{S}}^{\circledast}(t, \beta)\right]\right\}\end{array}\right.\)
- Properties
- Initial time: \(E_{\text {int }}(0):=\operatorname{Tr}\left[\rho_{\mathrm{S}}(0) H_{\mathrm{S}}\right] \quad\left(H_{\mathrm{S}}^{\circledast}(0, \beta)=H_{\mathrm{S}}\right)\)
TODO Compare with Rivas Method
DONE Find Rivas Paper
Rivas Vortrag
Matrix Eigenvals
- see cite:Pan1999May