Ultra-strong coupling simulation

[ustc...@gmail.com]

Hello everyone,

I am trying to study the ultra-strong light-matter interaction (light-matter polariton), in which the coupling strength becomes comparable to the photon frequency. In this regime, one  has to replace the Jaynes–Cummings model with the Rabi model, and thus the ground state involves virtual photons. An introduction to ultra-strong coupling can be found here:

Strong coupling tutorial

It is said that in the ultra-strong coupling regime, the normal master equation does not hold and one has to transform the basis to the photon-atom dresses state basis, which seems to make the modeling in QuTiP much more involved. A detailed explaination is given in the following paper (page 3-4):

2011-PRA-Dissipation and ultrastrong coupling in circuit QED

My question is:

1. Is it possible to model the ultra-strong coupling regime straightforwardly in QuTiP,  without too much deriviation and transformation that has to be done by hand. If so, it will become much more convenient.
2. If QuTiP cannot do it  straightforwardly, how should I construct and input the Hamiltonian and Liouvillians in the dressed state representation in QuTiP? Or if the basis set transformation can be done easity in QuTiP?
3. I would be very grateful if anyone could offer me more suggestions, references, or best computer codes that will help modeling the ultra-strong coupling.

Many thanks in advance!

[Diego Alejandro Rasero Causil]

Kind regards, 

on the QuTip website is the following example.  This one can help you. 

https://nbviewer.ipython.org/github/jrjohansson/qutip-lectures/blob/master/Lecture-3B-Jaynes-Cumming-model-with-ultrastrong-coupling.ipynb

Enviado desde mi Huawei de Claro.

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De: "ustc...@gmail.com" <ustc...@gmail.com>
Fecha: mar., 2 ago. 2022, 6:27 a. m.
Para: "QuTiP: Quantum Toolbox in Python" <qu...@googlegroups.com>
Asunto: Ultra-strong coupling simulation

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[Diego Alejandro Rasero Causil]

You can also see this

https://nbviewer.ipython.org/github/qutip/qutip-notebooks/blob/master/examples/ultrastrong-coupling-groundstate.ipynb

[ustc...@gmail.com]

Dear Sir Enviado desde mi Huawei de Claro,

Many thanks for you kind and instant replies. After seeing you suggestions, I have learned these official examples with care. However, based on the above PRA paper, I still believe that the master equation used here is likely improper to describe the dissipative ultra-strong coupling system, the reason being follows (stolen from the PRA paper): 2011-PRA-Dissipation and ultrastrong coupling in circuit QED

"In the ultrastrong coupling regime however, |g,0> is no longer the ground state ... Therefore, even at T = 0, in which case no energy should be added to the system, relaxation will generate photons in excess to those already present in the ground state."  (In other word, the Liouvillians L[σ-] or L[a] which normally decay the system to the ground state, now pump the system! Since |g,0> is no longer the ground state. Instead, one should use other annihilation operaters in the dressed state representation.)

In the official examples, they use the following solver for dissipative evolution

mesolve(H, psi0, tlist, [sqrt(kappa) * a], [a.dag() * a, sm.dag() * sm])

which might be inproper in my personal opinion. Nevertheless, the non-dissipative descriptions should be fine.

As to the CORRECT modeling of the ultra-strong coupling, I think one might has to use the dressed-state representation. In other word, a change of basis set and defination of new operators might be needed. One example is shown in the PRA paper. 

Not sure if I am right and any suggestions will be highly appreciated.

Best Regards,

G. Chen

[Neill Lambert]

Dear Chen,

The bloch-redfield solver  works in the eigenbasis of the hamiltonian, and essentially can reproduce the type of results shown in that paper. There is an example for USC cavity-QED in the brmesolve notebook, which was recently updated by christian

https://github.com/qutip/qutip-tutorials/blob/main/tutorials-v4/brmesolve-cavity-QED.md

For a bit more control on what goes into your model you can also construct this type of master equation manually, using the eigenstates of your hamiltonian and mesolve.   I am attaching a notebook showing a comparison of that approach to the bloch-redfield solver and to the 'wrong' master equation which I think also reproduces figure 1 from that paper (though I didnt look at the notebook in detail for a while, so it may not be completely equivalent, or even correct, and should be taken with a grain of salt, but might be useful as  a starting point).

all the best

neill 

To view this discussion on the web visit https://groups.google.com/d/msgid/qutip/e76ebb7b-b4b0-4992-8225-c6f5311f6536n%40googlegroups.com.

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[ustc...@gmail.com]

Dear Prof. Lambert,

Firstly, my great thanks for your helpful reply! I have read your suggestions and evaluated the codes and it seems that I can reproduce Fig. 1 in the PRA paper. However, more time is required for me to digest these treatments and to be able to calculate other quantities.

In the current stage, I have been a little pullzed by the two different treatments in your code, where the same results were reached. On the one hand, you use the Lindblad solver (steadystate() function, am I right?) but with newly defined collapse operators (in the Hamiltonian eigenstate basis, similar to the 2011 PRA paper). On the other hand,  you directly use the Redfield equation solver (brmsolve) without basis transformation. Dose it mean that brmsolve can always be used disregarding whether it is in the weak or deep-strong coupling regime? Moreover, I think whichever solver one use, the physical quantities has to be evaluated in the eigenstate basis (which is physical), is that right?

Another question is that, in your opinion which solver (Lindblad or Redfield) is more convenient in the following situations:

1. Evaluation of various quantities such as phonon number and photon correlation function.
2. Studying more complex systems, such as the exciton-phonon coupling (polariton chemistry)?

Thanks again for you kind help.

Best regards,

G. Chen

[Neill Lambert]

Dear chen,

In the current stage, I have been a little pullzed by the two different treatments in your code, where the same results were reached. On the one hand, you use the Lindblad solver (steadystate() function, am I right?) but with newly defined collapse operators (in the Hamiltonian eigenstate basis, similar to the 2011 PRA paper). On the other hand,  you directly use the Redfield equation solver (brmsolve) without basis transformation. Dose it mean that brmsolve can always be used disregarding whether it is in the weak or deep-strong coupling regime? Moreover, I think whichever solver one use, the physical quantities has to be evaluated in the eigenstate basis (which is physical), is that right?

Brmsolve internally calculates the eigenstates and constructs the appropriate rate equation (for the precise description of what it is doing its better to look at the documentation). It is much more general than that manual lindblad example I made, and can also deal with degeneracies, whereas that lindblad example doesn't. Still, I think bloch-redfield might not always segue perfectly to the weak coupling master equation (it always calculates the eigenstates and constructs rates based on those, so only actually becomes equivalent to what you would call the weak-coupling master equation at zero coupling)... I have found using the non-secular option, use_secular = False, for the bloch-redfield solver overcomes some issues in that limit. However, this is quite a tricky point, and something you have to be careful with. There's some interesting recent literature on it, see https://arxiv.org/abs/1906.08893 for example.

 

For your second question,in general yes, if you are looking at quantities like cavity emission and so on you have to evaluate them in the eigenstate basis to get 'physical' results.   Just fyi, in their default form both solvers return a density operator in terms of the ''bare-basis''.

 

Another question is that, in your opinion which solver (Lindblad or Redfield) is more convenient in the following situations:

1. Evaluation of various quantities such as phonon number and photon correlation function.

They are both convenient for this, the bloch-redfield solver is just a little more 'black box' and numerically optimized.

Also, if one has a slowly driven hamiltonian, the bloch-redfield solver is quite optimized for that situation, as it requires diagonalizing the hamiltonian at each time step.

 

1. Studying more complex systems, such as the exciton-phonon coupling (polariton chemistry)?

As above, if you understand its range of application the bloch-redfield solver does a lot of leg-work for you (see https://github.com/qutip/qutip-tutorials/blob/main/tutorials-v4/brmesolve_phonon_interaction.md  for a more complex example).

If you want to go beyond weak-system-bath-coupling approximations, we recently added the HEOM solver to qutip which might be useful in some situations.

thanks

neill

[ustc...@gmail.com]

Dear Prof. Lambert,

Thanks very much for your patient explanations, they are really very helpful to me!

Best regards,

G. Chen