HEOMSolver and time-dependent hamiltonians

[Ricardo Afonso (Jeremias Geremias)]

Dear Team,

I would like to express my sincere gratitude for your outstanding contribution to the development of QuTiP and the extensive array of examples that have been made available to the community. Your work is invaluable to our research endeavors.

I am reaching out to seek assistance regarding the use of HEOMSolver in studying a specific quantum problem. I must preface my queries by acknowledging my limited expertise in utilizing QuTiP, and the possibility of my unfamiliarity with its intricate functionalities.

My colleagues and I have been diligently exploring the spin-boson model, specifically under the impact of an external time-dependent function in the bias term. This function manifests in various forms, including sine, cosine, and step-function, among others. Our objective is to garner numerical insight to guide the trajectory of our ongoing calculations. Our focus is predominantly centered on the regime of Ohmic baths under finite temperature and coupling scale, and we are also examining the underdamped regime. The flexibility to scrutinize diverse scenarios for our system is crucial, given the time-dependent Hamiltonian in our research.

In light of the above, I would like to pose the following questions:

1) Does the HEOMSolver possess the capability to solve the spin-boson model, taking into account the time-dependent Hamiltonian of the system?

2) In the affirmative, I have observed that it is feasible to model the bath for the solver. Would it be possible to provide a general explanation and pertinent references regarding the operation of this class? I have yet to gain comprehensive insight into this aspect.

Your assistance and insights would be greatly beneficial in advancing our research. Thank you in anticipation for your time and cooperation.

[Paul Menczel]

Dear Ricardo,

We are happy to see that people are interested in the qutip heom implementation! Thank you for reaching out with your questions.

HEOMSolver is capable to deal with a time-dependent system Hamiltonian. To use a time-dependent Hamiltonian, simply pass a 'QObjEvo' as the first parameter to the constructor of HEOMSolver.

QObjEvo is qutip's way of representing time-dependent quantum objects. On the qutip tutorials website, you can find an introduction to QObjEvo as well as some heom example notebooks. The example notebook 4 ("Dynamical decoupling of a non-Markovian environment") uses a time-dependent Hamiltonian as follows:

> H_d = qutip.QobjEvo([H_sys, [H_drive, drive_fast]])
More details about this example can be found in the accompanying paper in section IIF: https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.5.013181

In that paper and in the tutorials, you will also find examples on how to use the bath classes. Currently, overdamped Drude-Lorentz as well as underdamped spectral densities are supported "out of the box". For other types of baths, you can create a representation of the bath by fitting either the spectral density or the auto-correlation function of the bath. The procedure is shown (and the various fitting approaches compared) for the example of an Ohmic bath in the paper (section IIG) and in the heom tutorial notebook 1d. We are currently working on making this a bit more convenient for users.

Kind regards,

Paul