#+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]]
*** TODO 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
**** TODO Test it with the two-qubit model
**** TODO 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)
***** NEXT Adjust normalization of model
***** TODO Verify that this works
**** TODO 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 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
*** TODO Physical Implication Single Bath
- how far away from thermal state
- exponential decay for markov case?
*** TODO Think about Higher moments
*** TODO 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
*** TODO Two Qubits
**** NEXT Hamiltonian
- [[file:calca/qubit_model/general_model.xopp][see notes]]
- look at cite:Kato2015Aug
- 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
**** TODO Sweep
***** TODO Automatic Convergence Testing
***** TODO Steady State Detector
***** TODO Sweep Parameter Extremes
***** TODO Observables
****** TODO Flow Magnitude Modulation
****** Local Energy Gradient
- upper limit (in suitable units)
****** Orientation
****** Level Spacing
****** Coupling
****** BCF
****** TODO Entanglement
- dependence on flow and all of the above
- can any state be reached?
- unavoidable entanglements
- cite:Xu2020Sep zz coupling breaks entanglement
****** Rectification
- see cite:Micadei2019Jun for experiment
  - energy flow between two qubits
****** TODO "Classical states"?
- cite:Aurell2019Apr -> jump processes, one bath
- effective description
- rate/kinetic equations
*** TODO 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
*** TODO Realistic Models
- ask Kimmo about quantum dots
- look at prof. strunzs paper again
** DONE Talk
*** DONE Plan
**** RESOLVED How much introduction
*** DONE Figures
*** DONE TeX
** 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
**** TODO Truncation scheme
- what does it mean if the norms are small?
- apparently with coupling it still works
- maybe dynamic truncation
**** TODO 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
*** TODO Basic Results
**** Initial Slip
*** TODO Analytical Comparison
*** TODO Numerical Results
**** TODO One Bath
***** TODO Qubit
- convergence:
  - sample count
  - hierarchy depth
- initial slip dependence on BCF, coupling
- non hermitian coupling and nonzero temperature
- estimate of interaction energy
- phenomenology
- consitency

***** TODO Qutrid
- demonstration of non-thermal state

* Brainstorm/Ideas
** test convergence properly
** Compare with Rivas Method
**  classical/markov limit
** Relation between coerrelation time and hops depth
** Importance sampling for initial $z$
** Manifold trajectories
** BEC bath as realistic model
** Temperature Probe
** Rectifier
** Motor
*** Looking at what the interaction energy does: maybe even analytically.
*** Thermal Operations
** Entropy Dynamics
** Effective thermal states (forget coherences)
*** ASK what is eigenstate thermalization

*** Preferred Basis
** Automatic definition of interaction so that interaction energy stays zero
- control to generate a thermal operation
- is this possible
- (i think so in hops ;P)
** [[https://en.wikipedia.org/wiki/Jarzynski_equality][Jarzynksi Equality]]
- related to work on the total system
** 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?
* Questions
** RESOLVED what is a kinetic equation
** ASK 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?]]