Time dependent dissipation and Hamiltonian

[Jake Iles-Smith]

Hi,

I'm trying to calculate the dynamics of a TLS under pulsed excitation in contact with a thermal reservoir. For this I need both a time dependent Hamiltonian H_s(t) = W(t)sigmax(), and a dissipator with time dependent rates (i.e. the rates are a function of the pulse W(t)) that cannot be written in Lindblad form.  

It seems like the current methods in mesolve require one to write the dissipator in Lindblad form, does anybody know of alternative ways of coding this?

Cheers

Jake

[Kevin Fischer]

Hi Jake,

It sounds like since you only require time-dependent rates I think you'll be able to make use of the collapse operator time-dependent API (http://qutip.org/docs/3.1.0/guide/dynamics/dynamics-time.html).

For instance, you could use

H = [[sigmax(), W(t)]]

and 

c_ops = [[A, W(t)]]

where A is the operator you want to have a time-dependent rate for.

Kevin

[Jake Iles-Smith]

Hi Kevin,

Sorry for the slow(!) reply, unfortunately my master equation cannot be written in a Lindblad form (the secular approximation breaks down in the regimes of interest). I was curious whether one could use mesolve with a time-dependent Liouvillian of arbitrary form?

Cheers

Jake

[Paul Nation]

If you can wait a week, then you can use the new time-dependent Bloch-Redfield solver.

-P

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[Jake Iles-Smith]

Hi Paul

I can certainly wait! Out of interest, will the red field solver be able to deal with any system-environment interaction or at least be hacked to do so? I would like to eventually develop a master equation for a Polaron theory interaction Hamiltonian to describe phonons in QDs.

Cheers

Jake

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[Paul Nation]

The coupling to the environment is through a Hermitian operator.  So if you have a proper operator, then your good to go.  It will also at first be only for.the string based time dependence, as it is very computationally intensive.

-P

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[Kevin Fischer]

Hi Jake,

I'm currently putting together a paper that will discuss the Bloch-Redfield master equation for phonon-QD interaction. Though I don't really know how you could use the Bloch-Redfield equations in the polaron frame, maybe I'm not being creative enough :) Also, they're actually not identical for short times to the full convolutional Polaron theory (NZ form) since Bloch-Redfield requires a coarse-graining on short timescales while the NZ form does not. But maybe this isn't necessary, you can see the type of results I got for the Bloch-Redfield application of QD-phonon interaction in https://journals.aps.org/prb/abstract/10.1103/PhysRevB.90.241404, maybe it's sufficient for you. After Paul finishes the new solver I'm going to add a notebook on how I did these calculations, which I'd be happy to share with you imminently if that would be helpful.

Regarding direct implementation of the polaron equation, there's not a super easy way to do this in QuTiP. But I did do this once by taking a piece-wise Hamiltonian (and hence Liouvillian). For each time step, compute the Liouvillian from the time convolutional integral (with calls to the QuTiP propagator to compute the propagators in the integrals). Then call mesolve to evolve the density matrix one time step. Repeat.

Hope this is helpful.

Kevin

[Kevin Fischer]

Hi Jake,

I also just wanted to comment that I enjoyed your paper "Phonon limit to simultaneous near-unity efficiency and indistinguishability in semiconductor single photon sources." It's certainly possible to add arbitrary time-dependent Liouvillians to the master equation solver, but it's not a feature currently. Would definitely be something worth adding.

Kevin

[Jake Iles-Smith]

Hi Kevin,

Thanks for your reply, and I'm glad to here you enjoyed our paper!

I think that one would have to reformulate the Bloch-Redfield approach in the polaron frame --  though in principle the procedure to construct the 'polaronic tensor' is exactly the same, the correlation functions look quite different, and the interaction Hamiltonian takes a slightly different form. It's an interesting point regarding your piece-wise implementation of the polaron master equation though, I will have a think whether this is suitable for my needs.

Cheers

Jake