master-thesis/project.org

#+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]]
** Quantum Thermo
see [[id:2dbc6bb9-69b5-44a6-9136-71e2f1490703][Quantum Thermodynamics]]
- [[id:eb435d2d-2625-4219-ae18-224eba0fa8a4][Coherent States]]

* 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
** TODO 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]]
*** DONE TeX notes
- done with nonlinear
*** DONE verify that second hops state vanishes
*** DONE Adapt New HOPS
- [[file:python/energy_flow_proper/01_zero_temperature/notebook.org][Zero Temperature Checks out]]
- stocproc can generate the time derivative with fft
**** Finite Temperture
- [[file:python/energy_flow_proper/02_finite_temperature/notebook.org][seems to work]]
- except for a small drift in the integrated energy
- i tried lowering the temperature, no dice
- some weird canellation?
*** DONE Time Derivative in stocproc
- done for fft
*** DONE 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
- [[file:calca/heat_flow/nonzero_t_no_time_derivative.xoj][one can get around the time derivative]]
- i have implemented finite temperature [[file:python/richard_hops/energy_flow_thermal.org][here]]
**** DONE Think about transform
*** 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
*** DONE Test Nonlinear hops
- see [[file:python/richard_hops/energy_flow_nonlinear.org][here]]
*** DONE Generalize to two Baths
- bath-bath correlations -> none yet
**** DONE Implement HOPSFlow for multiple baths
**** DONE TeX the multibath
**** DONE TeX interaction energy
**** DONE Implement interaction energy for multiple baths.
- plot it for tal
**** DONE Test it with the two-qubit model
**** DONE Initial Slip
- [[file:calca/heat_flow/initial_slip_zero_int.xopp][see notes on zero interaction]]
- for self adj -> apparently tempertature independent
- gives good estimate of interaction energy order of magnitude ->
  proportional to integral of imag part of BCF -> normalizing to one
  is helpful: explains why ω_c has influence on coupling strength (as
  seen in the new trunc scheme)
***** DONE Adjust normalization of model
***** DONE Verify that this works
***** DONE Verify time dependent
- done in [[file:python/energy_flow_proper/08_dynamic_one_bath/coupling_modulation.org][here]]
***** DONE Tex It
**** HOLD Q-Trid -> how non-thermal?
**** DONE Influence ω_c on initial slip and shape
- see [[file:calca/heat_flow/initial_slip_zero_int.xopp][the notes]]
- without non-zero system: generally enhanced flow (why?)
*** TODO Analytic Verification
**** Valentin's QMB Gaussian states
***** DONE One Bath
- [[file:calca/heat_flow/gaussian_model.xoj][gaussian model]] (raw) and [[file:tex/gaussian_model/build/default/default.pdf][as pdf]]
- [[file:python/energy_flow_proper/03_gaussian/comparison_with_hops.org][hops consistent in zero temperature]]
- [[file:python/energy_flow_proper/04_gaussian_nonzero/comparison_with_hops.org][and nonzero temperature case]]

***** Two Baths
- [[file:calca/heat_flow/two_ho.xopp][straight generalization]] (raw) and [[file:tex/gaussian_model/build/default/default.pdf][as pdf]]
- seems to check out with [[file:python/energy_flow_proper/05_gaussian_two_baths/comparison_with_hops.org][HOPS]]
- analytic solution may have numeric instabilities
- ok: seems to be very susceptible to the quality of the BCF fit
- got it to work :)
  - mistake in formula
  - root quality
  - hops truncation

****** DONE Heat Flow Numerics
- sill issues with gaussflow
- root precision!
- fit quality
- switched to fitting 2/3 where bcf is big and the rest on the tail

****** TODO Port to new system
****** TODO Try less symmetric

*** 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...]]
*** DONE rivas 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: \(E_{\text {int }}(0):=\operatorname{Tr}\left[\rho_{\mathrm{S}}(0) H_{\mathrm{S}}\right] \quad\left(H_{\mathrm{S}}^{\circledast}(0, \beta)=H_{\mathrm{S}}\right)\)
**** DONE Find Rivas Paper
*** HOLD Physical Implication Single Bath
- how far away from thermal state
- exponential decay for markov case?
*** TODO Think about Higher moments
- see [[file:calca/heat_flow/higher_order.xopp][notes]]
*** HOLD Why does the expression containing the first hier. states converging faster.
** HOLD Steady State Methods
- [[file:python/energy_flow_proper/05_gaussian_two_baths/longhopsidea.org][cholesky transform]] seems to provide us with the posibility of
  generating tree like processes
- related to fubini
- may help improving steady state statistics
- see cite:Pan1999May

*** HOLD implement tree method
*** HOLD Think about eigenstates and dividing out the hamiltonian

** TODO Applications
*** TODO Prior Art
- cite:Kato2015Aug two qubits, two baths
- cite:Aurell2019Apr one qubit, two baths, analytical
- cite:Wang2021Jan one phonon mode + qubit, two baths, analytical, weak bath int
  - negative thermal conductance at low coupling strenght between
    qubit and mode
  - thermal transistor with two qubits and one mode

- cite:Kato2016Dec non-pertubative three-level, HEOM
- cite:Esposito2015Dec interaction energy break second law
- cite:Strasberg2021Aug new entropy
*** HOLD Two Qubits
**** NEXT Hamiltonian
- [[file:calca/qubit_model/general_model.xopp][see notes]]
- look at cite:Kato2015Aug
- cite:Kato2016Dec: nontrivial effects if bath couplings don't commute
- cite:Aurell2019Apr uses one qubit between two baths
  - spin boson like
- cite:Hita-Perez2021Nov Effective hamiltonians for two flux qubits
  - simplest form $J_{xx}$ coupling
  - gives physical parameter ranges
- cite:Hita-Perez2021Aug strong coupling of flux qubit to resonators
  - again derivation of effective hamiltonian
  - no +- couplings
- cite:Wang2021Jan
  - $\sigma_x$ coupling to bath
- cite:MacQuarrie2020Sep
  - zz interaction: capacitve interaction between charge qubits
- cite:Andersen2017Feb strong coupling to mode -> x coupling, transmon
- cite:Mezzacapo2014Jul effective transmon coupling xx
- maybe dephasing coupling to minimize effects
***** General Model
- lock z and y axis
- coupling most general without using identities (-> without modifying
  local hamiltonian)
- normalization of energy scales
- maybe use [[id:c7a6d61e-7d0f-4504-acab-f1971f58ee20][Specht's Theorem]] to test if the hamiltonians are unitarily related.
  - I've used a sufficient criterion. but maybe this is not necessary in the end

- [[https://github.com/vale981/two_qubit_model][implemented model generator and utilities]]
  - with automatic hops config generation
***** NEXT First Experiment
- use z coupling to bath and modulate coupling between qubits
- find good parameters for convergence
- ok that worked. nothing unexpected: see [[file:python/energy_flow_proper/06_two_qubit_first_experiments/zz_xx_test.org][the notebook]]
***** TODO TeX It :P
*** HOLD Three Bath Fridge

here is the paper I had in mind when we talked about the three-bath fridge.

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.108.070604


I don't know if this scenario has been considered in a strong coupling framework.


This fridge is working continuously. Maybe for HOPS a stroke-based model could be better to avoid long propagation to the steady state. Just as an example, here is an Otto-Fridge with strong coupling (I have not thou thoroughly read this paper)

https://link.springer.com/article/10.1140%2Fepjs%2Fs11734-021-00094-0

- cite:Karimi2016Nov -> one HO and two resonators
- cite:Mu2017Dec, cite:Binder2018 -> linear additive coupling can't be used to attain cooling
*** HOLD Realistic Models
- ask Kimmo about quantum dots
- look at prof. strunzs paper again
*** TODO Heat Engines
See cite:Binder2018.
- our strengths lie in medium/fast non-periodic driving
- carnot maybe good idea: expansion and coupling at the same time
- we need at least two baths -> non passive
- stronger coupling + coherence should decrease
- interesting effects if H(t) does not commute for different times
- adiabaticity still present even with stronger coupling?
- monotonic convergence to steady state is guaranteed cite:Feldmann2004Oct
  - distance measure is the relative entropy: not symmetric
- shortcut to adiabaticity -> performance boost
**** Ref 92
- convergence to limit cycle only for weak?
- I don't think so
**** TODO Look at 105 in cite:Binder2018
**** TODO Chapter Two: How applicable to our case?
**** Single Bath Time Dependence
- no unilimited energy extraction due to passivity
  - i thought: WRONG!, indeed you can, but it's likely bounded
  - N - times the same HO definitely is, see [[file:python/energy_flow_proper/ergo_stuff/ergotropy_bath_qubit.org][my ergotropy experiments]] and [[file:calca/qubit_model/passive_states_once_more.xopp][calculations]]
    - small but finite changes let things blow up. i suspect this was a waste of time
    - actually they don't, my numerics do not reach far enough
  - it is bounded: cite:Biswas2022May
- see also [[file:calca/heat_flow/initial_slip_zero_int.xopp][my notes on pure dephasing]] -> no energy transfer dephasing at all
- see [[file:python/energy_flow_proper/08_dynamic_one_bath/coupling_modulation.org][modulation experiments]] and cite:Binder2018
  - as far away from dephasing as is possible
- see cite:Biswas2022May for absolute limit
  - conicides with my calculations in the limit ω -> 0
***** TODO verify ergo inequality
***** TODO Tex It

**** TODO Connection to Prior Art
- find out how much theorems are violated
- are there STIRUP-like surprises: overlapping and swapping stages
***** TODO Find results to reproduce
- strong coupling with HO WM: cite:Wiedmann2021Jun
- stirling: non-markovian cite:Raja2021Mar
  - strokes separate, no overlap
  - apparently higher eff than quasistat -> but only without thermalization
  - only qubits
  - second order in coupling -> born approx, no bath change cite:Kofman2004Sep
- carnot-like: cite:Scopa2018Jun uses GKSL-Floquet
- qutrid, store ergotropy: cite:Santos2021Jun
  - markov :)
****** (old) spin-1/2 in weak-coupling: cite:Geva1992Feb
  - refers to laser with semigroup model: Curzon-Ahlborn efficiency (in classical limit)
    - speaks of endoreversibility
    - irreverisibility through coupling
  - this work: more easily compared with classical, b.c. no simultaneous heat contact
  - qubit: no classical analog, simple
  - questions: curzon-ahlborn still valid, approaching equilibrium
    limit?, effect of quantum mechanics per-se

******* Model
- **many** non interacting spins as working fluid (multiply everything by N)
  - **does this make a difference?**
- carnot cycle: two isothermal br., two adiabatic
- modulation has no zero, simpliy magnitude of magnetic field, commutes with \(H\)
  - effecive diagonality

******* Work, Heat, Temp
- power and heat naively defined by instantaneous limits
  #+DOWNLOADED: screenshot @ 2022-05-09 15:22:34
  [[file:Tasks/2022-05-09_15-22-34_screenshot.png]]

#+DOWNLOADED: screenshot @ 2022-05-09 15:22:54
[[file:Tasks/2022-05-09_15-22-54_screenshot.png]]
- cite:Binder2018 says this is problematic outside the limit cycle if
  modulation is fast: work vs. internal energy (do we have this problem?)
- Modulating H does not change population
- negative Temperatures as artifact of non-positive
******* Cycles
- temperature equilibration is performed
- sudden limit: otto cycle efficiency upper bound for all
- step cycle converges onto reversible
- final cycle: detailed balance for the gksl -> time dependent coefficients (but ok if slow-varying)
  otherwise problematic
- non-equilibrium -> "temperatures of the working fluid not the same as the baths"
******* Striking Findings
- different heat transfer law
- high temperature limit:
  - times for isothermal branches
  - at maximum power: times independent of the isotherm temperatures
    - explicit modulation
  - maximum power at curzon-ahlborn eff, effectiveness 1/2
  - similar to newton but need not be close to eq.
****** General Notions in cite:Kurizki2021Dec
- continous, article cite:Mukherjee2020Jan
******* Reciprocating Engines
- adiabatic limit: wm state diagonal, efficiency 1-ω_c/ω_h
- coherence generated when hamiltonian (system driving) does not
  commute with itself: extra (external) work
  - making the state non-passive is costing work
- in sudden limit: cohorence gives work extraciton, *markov*
  - non-passivity for unitary extraction from the work medium
  - all engine types are equivalent (map over one cycle) when action small cite:Uzdin2015Sep
    - equivalence of map, but not state inside cycle
    - thermodynamic heat/power also converge to same
    - continous engines only extrac work from coherences
******* Chap 10: Anti-Zeno
- Zeno: frequent measurement slow down evolution
- Anti-Zeno: bath interaction accelleration by frequent measurement
  - more common
- effect of frequent measurement may be produced by unitary
  - frequent changes in the coupling

******* TODO 18, 22 -> ergotropy
- tighter bound p. 268 for entropy change
- 18: nonthermal baths are special and may perform work
- 22: nonpassivity of piston states -> work
  - maybe later: *implement machine proposed in HOPS*

******* Chap 20: Simultaneously Coupled Heat Machine
- *spectral separation*
- quantum advantage through *anti-zeno effect*

  #+begin_quote
  Remarkably, for modulation rates that fall within the non-Markovian
  regime, power boosts are induced by the anti-Zeno effect (AZE)
  (Chs. 10, 16). Such boosts signify quan- tum advantage over
  heat-machines that commonly operate in the Markovian regime, where
  the quantumness of the system–bath interaction plays no role.  The
  AZE-induced power boost stems from the time-energy uncertainty rela-
  tion in quantum mechanics, which may result in enhanced system–bath
  energy exchange for modulation periods comparable to the bath
  correlation time.
  #+end_quote

- std. σ_x coupling
- non markov ME til second order: see cite:Kofman2004Sep, cite:Raja2021Mar
- use floquet me
- markovian limit: *diagonal ρ*
- for separated spectra: simple expression for work and current
- speed limit for modulation
  \(\omega(t)=\omega_{\mathrm{a}}+\lambda \Delta \sin (\Delta t)\)
  $\Delta_{\mathrm{SL}}=\omega_{\mathrm{a}} \frac{T_{\mathrm{h}}-T_{\mathrm{c}}}{T_{\mathrm{c}}}$
  \[
  \Delta<\Delta_{\mathrm{SL}} \Longrightarrow \mathcal{J}_{\mathrm{c}}<0, \mathcal{J}_{\mathrm{h}}>0, \dot{W}<0
  \]
  \[
  \eta=\frac{\Delta}{\omega_{\mathrm{a}}+\Delta} \quad\left(\Delta \leq \Delta_{\mathrm{SL}}\right)
  \]
- maximal power for flat spectral density near energy exchange
  frequecny and very hot bath
  \(\Delta_{\max }=\frac{1}{2} \Delta_{\mathrm{SL}}, \quad \eta\left(\dot{W}_{\max }\right)=\frac{1-\frac{T_{\mathrm{c}}}{T_{\mathrm{h}}}}{1+\frac{T_{\mathrm{c}}}{T_{\mathrm{h}}}} \geq \eta_{\mathrm{CA}}\)
  \(\eta_{\mathrm{CA}}=1-\left(\frac{T_{\mathrm{c}}}{T_{\mathrm{h}}}\right)^{1 / 2}\)

  - non-markovian Anti-Zeno
    1. WM and Bath coupled over $n\gg 1$ modulation periods where the
      period is much shorter than the bath correlation + spectral separation
    2. decouple baths for a time longer than the bath correlation time
       to remove correlations
    - power boost for detuned baths
    - working medium attains diagonal form with rate equations (weak coupling)
  - zeno regeime if we don't turn off soon enough
    - no work extraction except when correlations large

******** TODO Work, Heat definition in chap 19
- on-off switching affects energy and ergotropy exchange
- spectral separation: intermittend coupling to only one of the two baths
  - non-overlapping harmonics -> effective otto cycle?
  - so that one bath gives, the other takes

    \(\begin{aligned} \mathcal{L}_{j, \pm q}(t) \rho=& \frac{P_{q}}{2}\left[G_{j}\left(\omega_{0} \pm q \Delta\right)\left(\left[a \rho, a^{\dagger}\right]+\left[a, \rho a^{\dagger}\right]\right)\right.\\ &\left.+G_{j}\left(-\omega_{0} \mp q \Delta\right)\left(\left[a^{\dagger} \rho, a\right]+\left[a^{\dagger}, \rho a\right]\right)\right] \end{aligned}\)

- non-markovian master equation for diagonal DM: needed when the
  coupling time in the order of the correlation time
- small modulation depth
- I don't understand (19.40) -> see p 375,378
  - leads with KMS condition to fast convergence to steady state


19.3 Model Parameters:
 - frictionless: interaction and system commute with themselves temporally
 - coupling modulation much slower than system
 - equidistant spectrum
 - spectral separation
   - see above
 - born approx
 - Pauli ME
 - *optimal: hybrid cycle, smooth strokes are best*
   - friction is regenerated by returning to passive state (shortcut)
 - no active friction: classical counterparts, *quantum coherence is
   neither essential nor advantageous for HE performance*
   - likely no quantum advantage in markovian

******** Generalizations
- modulating the coupling as well
- bigger system, non-equidistant spectrum
- non-commuting hamiltonians (temporal)

***** TODO Find Theorems to break
- quantum speed limit
- quantum friction:
  - how much does non-commutativity of the system impact
- stochastic cycles: efficiency limit cite:Binder2018
- symmetry of expansion and compression
- modulating the nature of the coupling may be interesting
- fast driving + overlap of strokes
- level of non-adiabaticity
- **how much is spohn violated**
  - very much
- **efficiency lower when temp gradient higher**
  - cite:Santos2021Jun
- ergotropy production
- dependence on cutoff
- limit-cycle: constant energy and entropy? (probably)
- fast modulation: more complicated "einschwingen", energy exchange
  with external source not to be neglected
- sudden limit->finite work? and adiabatic limit.
  (maybe even easier to define with finite memory)
- reversibility? how to define?
- *sudden limit*: equivalence of continous and stroke broken with a lot of memory?
  - may need big actions
  - coherence is explicitly needed
- detect signatures from cite:Uzdin2015Sep
  - *continous engines*: coherences are only source of work
  - defines a classical engineu
- cite:Kurizki2021Dec: p. 268 -> heat and entropy inequalities may be
  broken, gives concrete conditions

- non-abrubt on-off, seems to be a problem for cite:Kurizki2021Dec
- noncommuting coupling to the two baths

- cite:Strasberg2021Aug mutual information large, correlations
  diminish when system is driven
***** TODO Model Ideas
- for starters: qubit
- two coupled qubits also nice
- non-scalar time dependence
- period of high int-strength followed by period of low for thermalization
- maybe extra dephasing step -> should remove power output
- notion of instantaneous temperature? cite:Geva1992Feb
- spectral separation
- time-scales in the order of bath correlation times or shorter
- continous cycle machines: may have quantum advantage cite:Kurizki2021Dec
  - coherence work extraction
  - maybe contrast stroke vs continous?

- later: three level system or two qubits
- crossover between otto and hybrid cycles
**** DONE Implement Two-Bath Qubit
- see my experiment: [[file:python/energy_flow_proper/10_antizeno_engine/anti_zeno_engine.org][anti zeno engine]]
  - initial results suggest, that there is indeed some finite time
    effect
  - spectral separation is important
  - detuning is important -> only then non-markov effects
**** TODO cite:Uzdin2015Sep repro of cite:Klatzow2019Mar
- transient effects missing
- deviations for long modulation periods, or large actions
**** TODO stroke based on coupling modulation, my energy shovel
- maybe even three level

** DONE Talk
*** DONE Plan
**** RESOLVED How much introduction
*** DONE Figures
*** DONE TeX
** TODO Poster
*** DONE Abstract
- motivation
- tanimura paper cite
- features/capabilities of our thing
**** DONE Figures
- Agreement with HO
- Consistency
- Energy shovel
- (maybe: anti-zeno)
*** TODO Structure
**** TODO Motivation
- similar to warb presentation
- mention HEOM result
**** TODO Main Result
- show main formula for flow and interaction
- nonlinear + finite temp + time dep -> most general
  - single bath for brevity
- mention possibility to calculate other B operator Quantities
**** TODO Analytical Verification
- show one and two bath plots
- short mention of the model and params
**** TODO Initial Slip
- show flow consistency for ω and δ dependence + initial slip
**** TODO Driven System
- show ω_c dependence of energy shovel
***** TODO Generate Good Plots + Precision
**** TODO Maybe: Anti Zeno Engine or Stroke Based
- plot power zeno vs anti-zeno
**** TODO Outlook
- more systematics: process mean vs hierarchy states
- convergence criteria
  - consistency between methods

** HOPS Numerics
*** DONE Stable Norm
- see [[file:calca/hops/auto_norm.xopp][notes]]
- already implemented
**** DONE TeX it
*** DONE Fock HOPS
- see [[file:calca/hops/fock_hops.xopp][notes]]
- already implemented
- intesting: anti-herm part is probability decay
- decay is stronger the higher the depth
**** DONE TeX it
**** HOLD Truncation scheme
- what does it mean if the norms are small?
- apparently with coupling it still works
- maybe dynamic truncation
**** DONE TeX It
** Quantum Thermo
*** How is heat flow measured?
- cite:Stevens2021Sep energy change in qubit drive field conditioned on measurement outcome
  - cites papers with engines fueled by measurements
** TODO Writing Up
*** TODO Intro
- recent interest in quantum thermodynamics
- no consensus: maybe for periodic steady state but not transient
- new tools required
- non markov: may be key? -> cite some papers, transient dynamics,
  non-eq
  - cite:Kato2016Dec shows that under strong coupling definitions can diverge
  - general dynamics interesting: mention settled weak coupling/markov case
- most methods -> manual access pertubative
  - cite kato papers cite:Kato2015Aug, cite:Kato2016Dec -> result for HEOM
  - HOPS can do this too (and likely better)
- HOPS side of motivation: we actually compute the whole unitary dynamics

*** TODO Short Mention of NMQSD and HOPS
- simple description, refer to appendix for details

*** TODO Basic Results
- how to calculate flow and interaction
- higher orders

**** TODO Cleaning it up
- proper chapters
- more prose

*** TODO Analytical Comparison
- brief review of the solution
- basic demonstration
- *maybe*: more numerics needed
- lessons learned

*** TODO Some Basic Quantum Thermodynamics
- operational results about ergotropy
- hint at next chapter
- support argument for bath memory on nmqsd and hops level
  - point to numerical result (somewhere in the 08 project...)
- cite:Lobejko2021Feb -> weird: locked energy in coherences ->
  restriction through thermal ops

**** TODO Explicit Calculation for Bath with "infinite Memory"
- N identical HO
- point out, that bound will be saturated if level spacing becomes
  continous (a conjecture!!!!)

*** TODO Numerical Results
**** TODO One Bath Thermo
***** TODO Model and Convergence
- model and bcf normalization
- convergence:
  - consistency check
  - sample count
  - stocproc
  - hierarchy depth
  - flow faster than system (sometimes)

***** TODO Initial Slip
- constant coupling
- dynamics coupling
- initial slip dependence on BCF, coupling, also for time dependent
- energy-transport requires going away from pure dephasing
  - somewhat "classical" in its nature

***** TODO Energy Reduction of the Bath
- show energy shovel
- compare with σ_+ coupling
- show friction vs frictionless
- show with system vs without
- show modulation frequency dep -> speed limit
- show detuning depency -> resonance effect
- show ω_c depence -> genuine non-markov features
- discuss steady state behavior in light of theory

****** TODO a truckload of cleanup in [[file:python/energy_flow_proper/08_dynamic_one_bath/coupling_modulation.org][here]]
****** TODO find out if there is an applicaple markov result

**** TODO Two Baths
***** TODO short demo of two qubits coupled to two baths
- mention significance of non-commuting coupling: cite:Kato2016Dec
***** TODO short demo of the otto cycle
- mention papers on the topic
  - mention curzon ahlborn and squeezed bath
- only demonstration, no systematic enquiry made
***** TODO Anti-Zeno Engine
- short mention of the paper and the idea
- nice because: non-markovian
- description of the model, frictionless dynamics
- explaination of choice of parameters
- show basic anti-zeno result
- mention coupling/decoupling effects

****** TODO maybe try two hot baths :P
****** TODO maybe calculate interaction fluctuation

***** TODO MAYBE try to make something out of the shovel

***** TODO Outlook
- mention interesting future project ideas
- cite:Kolar2012Aug -> quantum refrigerator, subohmic SD->coling to zero in finite time
- cite:Magazzu2018Apr comparison to experiment: driven spin boson


* Brainstorm/Ideas
** Initial Coherences -> more work extraction
** Spohn
** Weak coupling second law
** Test new entropy definition vs extracted work
** Non monotonous entropy propduction <-> increased output?
** Compare with Rivas Method
- especially in the light of the ergo inequality
** classical/markov limit
- high temperature
- delta correlations
** Importance sampling for initial $z$
** BEC bath as realistic model
** engines
- cite:Santos2021Jun
** Ergotropy
** Eigenstate Temperature
** cite:Esposito2015Dec exclude definitions because not exact differential
** What happens to the interaction H in steady state
** Why does everything come to a halt except the bath?
** ASK General Coupling Operators?
** Correlations between baths
* Questions
** RESOLVED what is a kinetic equation
** DONE what is feschbach projection
** DONE Look up Michele Campisi
- identify heat source first: then definition :)
- entropy production positive not quite second law: not thermodynamic entropy
  - stricter
** DONE Landauer Principle
** DONE Logical vs. Theromdynamic Irreversibility
- logical: no info is lost in computation

** RESEARCH [[id:c55b6bac-87e3-4b23-a238-c9135e3c1371][Quantum Fluctuation theorems?]]
** RESEARCH Do the enhancements in energy flow originate from the shift of the peak or from the absence of low energy modes?
* Problems
** Ray on slurm
- ray suddenly needs ~--include-dashboard False~