#+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~