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13 KiB
13 KiB
- Literature
- Tasks
- Implement Basic HOPS
- Quantify Heat Transfer
- Steady State Methods
- Applications
- HOPS Numerics
- Quantum Thermo
- 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
- Entropy Dynamics
- Effective thermal states (forget coherences)
- Automatic definition of interaction so that interaction energy stays zero
- Jarzynksi Equality
- engines
- 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?
- Questions
Literature
Stochastic Processes
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
HOPS
Numerik
Quantum Thermo
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
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
- my first experiments yield bogus numerics…
- richards code makes it work
-
for derivations see
- 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
DONE Adapt New HOPS
- Zero Temperature Checks out
- stocproc can generate the time derivative with fft
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
- see here
TODO Generalize to two Baths
- bath-bath correlations -> none yet
NEXT Implement HOPSFlow for multiple baths
NEXT TeX the multibath
NEXT TeX interaction energy
TODO Analytic Verification
- cummings
- and pseudo-mode
Valentin's QMB Gaussian states
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
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 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
TODO 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
TODO implement tree method
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
TODO 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
- upper limit (in suitable units)
- dependence on flow and all of the above
- can any state be reached?
- unavoidable entanglements
- cite:Xu2020Sep zz coupling breaks entanglement
-
see cite:Micadei2019Jun for experiment
- energy flow between two qubits
- cite:Aurell2019Apr -> jump processes, one bath
- effective description
- rate/kinetic equations
TODO 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
NEXT Realistic Models
- ask Kimmo about quantum dots
HOPS Numerics
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
TODO Truncation scheme
- what does it mean if the norms are small?
- apparently with coupling it still works
- maybe dynamic truncation
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
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)
ASK 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?
Questions
RESOLVED what is a kinetic equation
ASK 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