master-thesis/project.org

• Literature
  • Stochastic Processes
  • Open Systems
  • Stochastic Unravelings
  • NMQSD
  • HOPS
  • Numerik
• Important Basics
• Todo
  • RESEARCH Quantum Fluctuation theorems?
• Tasks
  • Implement Basic HOPS
  • Quantify Heat Transfer
    • TeX notes
    • verify that second hops state vanishes
    • NEXT Adapt New HOPS
    • Time Derivative in stocproc
    • Generalize to Nonzero Temp
      • Think about transform
    • Try to get Richards old HOPS working
    • Test Nonlinear hops
    • Generalize to two Baths
    • Analytic Verification
      • Valentin's QMB Gaussian states
    • figure out why means involving the stoch. process are so bad
    • VORTRAG
    • Compare with Rivas Method
    • Find Rivas Paper
    • NEXT Make proper library
    • Three Bath Fridge
      • Mail from Konstantin
  • Find the Steady State
  • Matrix Eigenvals
  • Relation between coerrelation time and hops depth
• Ideas

Literature

Stochastic Processes

• Introduction to stochastic processes-lecture notes
• Stochastic processes for physicists: understanding noisy systems
• Probability and stochastic processes for physicists ||

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

See also NMQSD.

• The non-markovian stochastic schr\ifmmodeo\else\"o\fidinger equation for open systems
• Non-Markovian Quantum State Diffusion
• Open system dynamics with non-markovian quantum trajectories

HOPS

See also HOPS.

• Hierarchy of stochastic pure states for open quantum system dynamics
• Exact open quantum system dynamics using the hierarchy of pure states (hops)
• Open quantum system response from the hierarchy of pure states

Numerik

See Numerics

• Numerical Recipes

Important Basics

• Coherent States

Todo

RESEARCH Quantum Fluctuation theorems?

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

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

  • nonlinear
  • TeXed notes
• 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

DONE verify that second hops state vanishes

NEXT Adapt New HOPS

• Zero Temperature Checks out
• stocproc can generate the time derivative with fft

TODO Time Derivative in stocproc

• done for fft

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 Think about transform

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 Analytic Verification

• cummings
• and pseudo-mode

TODO Valentin's QMB Gaussian states

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…

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

NEXT Make proper library

TODO Three Bath Fridge

Mail from Konstantin

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

Find the Steady State

Matrix Eigenvals

• see cite:Pan1999May

Relation between coerrelation time and hops depth

Ideas

• tune stocproc tansinh with fft