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Literature

Stochastic Processes

Open Systems

Stochastic Unravelings

NMQSD

See also NMQSD.

HOPS

See also HOPS.

Numerik

See Numerics

Quantum Thermo

see Quantum Thermodynamics

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

TODO Quantify Heat Transfer

DONE TeX notes

  • done with nonlinear

DONE verify that second hops state vanishes

DONE Adapt New HOPS

Finite Temperture
  • 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 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
DONE 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

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
DONE Test it with the two-qubit model
TODO Initial Slip
  • 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
TODO Verify time dependent
HOLD Q-Trid -> how non-thermal?
DONE Influence ω_c on initial slip and shape
  • see the notes
  • without non-zero system: generally enhanced flow (why?)

TODO Analytic Verification

Valentin's QMB Gaussian states
DONE One Bath
Two Baths
  • straight generalization (raw) and as pdf
  • seems to check out with 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

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

HOLD Why does the expression containing the first hier. states converging faster.

HOLD Steady State Methods

  • 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

HOLD Two Qubits

NEXT Hamiltonian
  • 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 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

  • 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 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

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
TODO Ref 92
  • convergence to limit cycle only for weak?
  • I don't think so
TODO Look at 105
TODO Chapter Two: How applicable to our case?
DONE Single Bath Time Dependence
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
(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

    /hiro/master-thesis/media/commit/dd22a26afabe09aad40ff59ce060b3699533eaad/Tasks/2022-05-09_15-22-34_screenshot.png

/hiro/master-thesis/media/commit/dd22a26afabe09aad40ff59ce060b3699533eaad/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
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
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
TODO Find Theorems to break
  • quantum speed limit
  • quantum friction
  • 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
  • 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

  • detect signatures from cite:Uzdin2015Sep

    • continous engines: coherences are only source of work
    • defines a classical engine
  • cite:Kurizki2021Dec: p. 268 -> heat and entropy inequalities may be broken, gives concrete conditions
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

  • continous cycle machines: may have quantum advantage cite:Kurizki2021Dec

    • coherence work extraction
    • maybe contrast stroke vs continous?
NEXT Implement Two-Bath Qubit

DONE Talk

DONE Plan

RESOLVED How much introduction

DONE Figures

DONE TeX

HOPS Numerics

DONE Stable Norm

  • see notes
  • already implemented
DONE TeX it

DONE Fock HOPS

  • see 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

TODO Hopsflow Power

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)

DONE 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)

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?

ASK General Coupling Operators?

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 Quantum Fluctuation theorems?