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[2021-10-07 Thu 13:38]--[2021-10-07 Thu 17:50] => 4:12 **** DONE TeX interaction energy **** DONE Implement interaction energy for multiple baths. - plot it for tal -**** TODO Test it with the two-qubit model +**** DONE 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 @@ -94,9 +94,10 @@ CLOCK: [2021-10-07 Thu 13:38]--[2021-10-07 Thu 17:50] => 4:12 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 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 [[file:calca/heat_flow/initial_slip_zero_int.xopp][the notes]] - without non-zero system: generally enhanced flow (why?) @@ -123,7 +124,8 @@ CLOCK: [2021-10-07 Thu 13:38]--[2021-10-07 Thu 17:50] => 4:12 - fit quality - switched to fitting 2/3 where bcf is big and the rest on the tail -****** TODO Try less symmetric +****** 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 @@ -142,11 +144,12 @@ CLOCK: [2021-10-07 Thu 13:38]--[2021-10-07 Thu 17:50] => 4:12 - 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 +*** HOLD 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. +- 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 @@ -165,7 +168,7 @@ CLOCK: [2021-10-07 Thu 13:38]--[2021-10-07 Thu 17:50] => 4:12 - negative thermal conductance at low coupling strenght between qubit and mode - thermal transistor with two qubits and one mode -*** TODO Two Qubits +*** HOLD Two Qubits **** NEXT Hamiltonian - [[file:calca/qubit_model/general_model.xopp][see notes]] - look at cite:Kato2015Aug @@ -223,7 +226,7 @@ CLOCK: [2021-10-07 Thu 13:38]--[2021-10-07 Thu 17:50] => 4:12 - cite:Aurell2019Apr -> jump processes, one bath - effective description - rate/kinetic equations -*** TODO Three Bath Fridge +*** HOLD Three Bath Fridge here is the paper I had in mind when we talked about the three-bath fridge. @@ -238,9 +241,146 @@ This fridge is working continuously. Maybe for HOPS a stroke-based model could b https://link.springer.com/article/10.1140%2Fepjs%2Fs11734-021-00094-0 - cite:Karimi2016Nov -> one HO and two resonators -*** TODO Realistic Models +- 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 +- no energy extraction due to passivity + - 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 + +- 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 + +**** 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 + #+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 +******* 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 @@ -257,11 +397,12 @@ https://link.springer.com/article/10.1140%2Fepjs%2Fs11734-021-00094-0 - intesting: anti-herm part is probability decay - decay is stronger the higher the depth **** DONE TeX it -**** TODO Truncation scheme +**** HOLD Truncation scheme - what does it mean if the norms are small? - apparently with coupling it still works - maybe dynamic truncation -**** TODO TeX It +**** 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 @@ -301,7 +442,7 @@ https://link.springer.com/article/10.1140%2Fepjs%2Fs11734-021-00094-0 *** Thermal Operations ** Entropy Dynamics ** Effective thermal states (forget coherences) -*** ASK what is eigenstate thermalization +*** DONE what is eigenstate thermalization *** Preferred Basis ** Automatic definition of interaction so that interaction energy stays zero @@ -317,9 +458,10 @@ https://link.springer.com/article/10.1140%2Fepjs%2Fs11734-021-00094-0 ** 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 -** ASK what is feschbach projection +** 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