#+STARTUP: content
#+FILETAGS: Uni Master

* Literature
** Stochastic Processes
- [[id:223952d2-a9fa-4c96-b429-f05fd08644ca][Introduction to stochastic processes-lecture notes]]
- [[id:80a1efbe-130e-4236-a5bc-a29dc81ea57a][Stochastic processes for physicists: understanding noisy systems]]
- [[id:8559e06e-8681-4fc6-86ff-5732aefacca7][Probability and stochastic processes for physicists ||]]
** Open Systems
- [[id:c2e028d9-7ba5-4bbe-8c45-b191c6001f9a][Open Quantum Systems]] by Rivas
- [[id:bbcfafbe-685a-4773-9391-119230199e67][Fundamentals of quantum optics benjamin]] by Klauder
** Stochastic Unravelings
- [[id:d1b1ff19-6450-48e5-96b7-cf0ba75e33d0][The quantum-state diffusion model applied to open systems]] one of the first applications
- [[id:487f7392-2db2-474d-a97d-2392b8801a58][Decoherent histories and quantum state diffusion]]
** NMQSD
See also [[id:0c2d1e58-7af7-411a-ace4-b6cc9e16859b][NMQSD]].
- [[id:f621ce90-bf29-4ee7-8972-618d41eb5092][The non-markovian stochastic schr\ifmmodeo\else\"o\fidinger equation for open systems]]
- [[id:abb3e07e-ce6f-4ab8-bc88-f00f80196ed6][Non-Markovian Quantum State Diffusion]]
- [[id:c3fc86bd-8b17-4015-b12d-b2a345da49c3][Open system dynamics with non-markovian quantum trajectories]]
** HOPS
See also [[id:ddb3a3ad-c876-461d-b634-4bb5d330e25a][HOPS]].
- [[id:d98cf8bd-ec91-42a7-bea9-1d196ed42c32][Hierarchy of stochastic pure states for open quantum system dynamics]]
- [[id:e5a44f45-2120-44ce-8e74-5ae247fa977e][Exact open quantum system dynamics using the hierarchy of pure states (hops)]]
- [[id:66e7eaf1-24a8-4a14-826e-1f132823fa9a][Open quantum system response from the hierarchy of pure states]]
** Numerik
See [[id:f8d8a28b-7ae3-425a-921e-8f472b166866][Numerics]]
- [[id:f056e38e-d46b-40c5-bc69-5a14d2db2c88][Numerical Recipes]]
* Important Basics
- [[id:eb435d2d-2625-4219-ae18-224eba0fa8a4][Coherent States]]

* Todo
** DONE Where is stochastic unraveling explained in more detail?
- maybe in sources 1-7 in the cite:Diosi1997
  1. 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 [[id:85fc22ad-ad87-4f6e-a395-00c6fb33f263][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?
** ASK [[id:c55b6bac-87e3-4b23-a238-c9135e3c1371][Quantum Fluctuation theorems?]]
** 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 [[file:calca/nmqsd_doodles/nonlin_heisenberg.xoj][see the end of my notes]]
* Tasks
** DONE Implement Basic HOPS
:LOGBOOK:
CLOCK: [2021-10-08 Fri 08:51]
CLOCK: [2021-10-07 Thu 13:38]--[2021-10-07 Thu 17:50] =>  4:12
:END:
- see [[file:python/experiments/stochproc/test_stoch.org][my stoch. proc experiments]]
- ill use [[https://github.com/cimatosa/stocproc/tree/master/stocproc][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
- [[file:python/billohops/test_billohops.org][my first experiments]] yield bogus numerics...
- richards code makes it work
- for derivations see
  - [[file:calca/heat_flow/nonlinear_hops.xoj][nonlinear]]
  - [[file:tex/energy_transfer/main.pdf][TeXed notes]]
- the energy balance checks out [[id:cbc95df0-609d-4b1f-a51d-ebca7b680ec7][System + Interaction Energy]] and [[file:calca/heat_flow/hsi.xoj][my notes]]
- i've generalized to multiple exponential in [[id:9ce93da8-d323-40ec-96a2-42ba184dc963][this document]]
*** TODO TeX notes
- done with nonlinear
*** TODO verify that second hops state vanishes
*** DONE Try to get Richards old HOPS working
- code downloaded from [[https://cloudstore.zih.tu-dresden.de/index.php/s/9sdcn3FGGbDMDoj][here]]
- it works see [[file:python/richard_hops/energy_flow.org][Energy Flow]]
- interestingly with this model: only one aux state
*** TODO Test Nonlinear hops
- see [[file:python/richard_hops/energy_flow_nonlinear.org][here]]
*** TODO Generalize to two Baths
*** 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 [[file:calca/heat_flow/thermal.xoj][notes]]
- includes time derivative of stoch proc
- idea: sample time derivative and integrate
- not as bad as thought: no exponential form needed -> process smooth
*** 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: [[id:2872b2db-5d3d-470d-8c35-94aca6925f14][Energy Flow in the linear case
  with smooth correlation...]]
**** ASK
- why do i have to take the conjugate of the process?
**** TODO get around that limitation
- maybe use integrated form
*** TODO 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
*** TODO Find Rivas Paper
** Rivas Vortrag
** Matrix Eigenvals
- see cite:Pan1999May
** Relation between coerrelation time and hops depth