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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

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
  • 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 Tex It
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
  • cite:Kato2016Dec non-pertubative three-level, HEOM
  • cite:Esposito2015Dec interaction energy break second law
  • cite:Strasberg2021Aug new entropy

HOLD Two Qubits

NEXT Hamiltonian
  • 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 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

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 my ergotropy experiments and 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 my notes on pure dephasing -> no energy transfer dephasing at all

  • see 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

    /hiro/master-thesis/media/commit/c5705512f4c20744d5e1e6ca3fc2f7af3e27fbb9/Tasks/2022-05-09_15-22-34_screenshot.png

/hiro/master-thesis/media/commit/c5705512f4c20744d5e1e6ca3fc2f7af3e27fbb9/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

    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.

  • 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: 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 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

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

  • mention coupling/decoupling effects
  • cite:Lobejko2021Feb -> weird: locked energy in coherences -> restriction through thermal ops
  • long strokes: no mixing in hierarchy -> trunc scheme
  • future diffrent coupling to same bath -> thermal machine (shouldn't be possible)
  • mention curzon ahlborn and squeezed bath
  • mention potential power consistency check

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

TODO Basic Results

  • how to calculate flow and interaction
TODO Maybe Higher Moments
  • at least hint at it -> cutoff

TODO Analytical Comparison

  • brief review of the solution
  • basic demonstration
  • maybe: more numerics needed
  • lessons learned

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 Energy Reduction of the Bath
  • energ

  • initial slip pure dephasing

    • consequences for design
    • estimate of interaction energy

  • ergotropy results: N identical HO, small detuning + consecutive interaction

    • general argument
  • initial slip dependence on BCF, coupling, also for time dependent
  • non hermitian coupling and nonzero temperature
  • fast driving of coupling or system or both

  • maybe theories to explain, probably resonance effects

    • look at golden rule etc
  • support argument for effective passive state by uncoupling-recoupling the bath
TODO Two Baths
TODO short demo of two qubits coupled to two baths
  • mention significance of non-commuting coupling: cite:Kato2016Dec
TODO Engines

Brainstorm/Ideas

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 Quantum Fluctuation theorems?