Expectation/Steady State of Time dependent operators

[Bharath Kannan]

I am trying to compute the Expectation values and/or steady state for a time dependent operator (my Hamiltonian is time dependent as well). Is there a way to do this in QuTiP?

[Paul Nation]

You would have to output the states from the solver that you want, and then compute the expectation value of your time-dependent expectation operator.  The steadystate solver assumes a time-independent system, so you will just have to evolve the system for long enough to reach steady state.

-P

Bharath Kannan

February 5, 2017 at 13:04

I am trying to compute the Expectation values and/or steady state for a time dependent operator (my Hamiltonian is time dependent as well). Is there a way to do this in QuTiP?

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[Bharath Kannan]

Would I construct the Time Dependent Operator in the same way I construct a Time Dependent Hamiltonian?

[Paul Nation]

You could.  How though doesn't matter as long as the output is a qobj object.

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[Kevin Fischer]

Hi Bharath,

There are three ways I'm aware of to compute a "steady state" for a time-dependent operator/Hamiltonian, more generally called a quasi-steady state since the density operator may be periodic in time.

1) Transform your Hamiltonian and operator to a rotating frame with no time-dependence, then you can use the steadystate solver.

2) Method of continued fractions which is a perturbative approach, for when (1) does not work (e.g. as in https://web.stanford.edu/group/nqp/jv_files/papers/alexander-bichromatic.pdf).

3) Floquet formalism which is an exact approach, for when (1) does not work (http://qutip.org/docs/4.0.2/guide/dynamics/dynamics-floquet.html). Though with a time-dependent operator QuTiP does not automatically allow you to compute an expectation value, so it would require a good understanding of Floquet theory.

Hope one of these works for your specific problem.

Kevin

[Pu ZHANG]

Hi, Kevin! 

I have a question related to your discussions here. 

My Hamiltonian is periodic in time and can't be transformed to time-independent. Following the User Guide, it is possible to evolve the system in time using mesolve. After propagating for some time, a "quasi-steady" state is seen reached. 

I would like to study the spectrum or other correlation properties of an operator, which is time-independent. I guess these couldn't be rigorously defined for a "quasi-steady" state, but is there something one might use to approximate these features? 

The Floquet formalism seems applicable to my problem (is collapse operator allowed in the formalism?), yet I'm not sure what more I can get from Floquet formalism than simple propagation in time. 

Thanks! 

Best regards, Pu Zhang

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Faculty at School of Physics, Huazhong University of Science and Technology

Room 819 (N.), Yifu Science and Technology Building

1037 Luoyu Road, Wuhan, China

E-mail: puzha...@hust.edu.cn

Homepage: www.researchgate.net/profile/Pu_Zhang4

Phone: +86 18871860394

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[Kevin Fischer]

Hi Pu,

If the Hamiltonian cannot be made time-independent through a single unitary transformation, then I believe the issue is that you don't always get a nice quasi-steady state. You might need an arbitrarily large time-series to describe what's happening, but the Floquet picture helps you figure out which terms are relevant. Most likely, what you would need to do is transform to a Floquet representation, apply a type of perturbation theory, and then transform back to generate an effective Hamiltonian. Here's a presentation that might serve as a good starting point for understanding Floquet theory and how it might apply to this problem.

You definitely can add collapse operators to Floquet master equations, but there's a bit of subtly in terms of what the floquet states mean depending on whether you derive the master equation in the Floquet picture or if you transform the master equation into the Floquet picture. Qutip's fmmesolve implements one of these, though, and the ordering doesn't matter since you're evaluating an operators expectation rather than interpreting the meaning of the Floquet states.

Additionally, that presentation discusses a technique called "average Hamiltonian theory" which seems like it also might be applicable and is potentially a bit easier to understand here.

Kevin

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[Pu ZHANG]

Thanks a lot for the reply! I will look into the info. 

Best! 

Best regards, Pu Zhang

--

Faculty at School of Physics, Huazhong University of Science and Technology

Room 819 (N.), Yifu Science and Technology Building

1037 Luoyu Road, Wuhan, China

E-mail: puzha...@hust.edu.cn

Homepage: www.researchgate.net/profile/Pu_Zhang4

Phone: +86 18871860394

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[Pu ZHANG]

Hi, Kevin! 

I have some follow-up questions from your kind comments. Regarding the function fmmesolve in QuTiP, it seems the current implementation only allows for one collapse operator, then problems involving more than one collapse operators couldn't be treated, right? Now I'm trying to study the behaviors by using mesolve with time-dependent Hamiltonian. From some expectation outputs as functions of time, it can be seen the dynamic is periodic (although not sinusoidal) after certain time's propagation. I think in such situation it is still possible to extract the emission spectrum of a cavity in the system. My concern is if I can do the spectrum calculation within QuTiP. It seems the built-in functions are not applicable for time-dependent Hamiltonians. Probably I can calculation correlation function somehow and do Fourier transform? Thanks a lot! 

Best! 

Best regards, Pu Zhang

--

Faculty at School of Physics, Huazhong University of Science and Technology

Room 819 (N.), Yifu Science and Technology Building

1037 Luoyu Road, Wuhan, China

E-mail: puzha...@hust.edu.cn

Homepage: www.researchgate.net/profile/Pu_Zhang4

Phone: +86 18871860394

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[Pu ZHANG]

I have another conceptual question as for Floquet formalism. Suppose the dynamics has periodic (or almost so) behavior after long time propagation. Does this imply that the system could be well approximated by an effective time-independent Hamiltonian? This effective Hamiltonian then can be found through Floquet formalism. Is such understanding correct? Thanks! 

Best regards, Pu Zhang

--

Faculty at School of Physics, Huazhong University of Science and Technology

Room 819 (N.), Yifu Science and Technology Building

1037 Luoyu Road, Wuhan, China

E-mail: puzha...@hust.edu.cn

Homepage: www.researchgate.net/profile/Pu_Zhang4

Phone: +86 18871860394

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[Kevin Fischer]

Hi Pu,

Although I haven't had to solve a problem like this before, I can try to offer a starting point. Even without Floquet theory, since your dynamics are periodic (say time T1), you can probably get the spectrum using mesolve to calculate a bunch of two-time correlations. If the two-time correlations are also periodic with a consistent period (say time T2), then I think can offer advice. My guess is that getting the spectrum requires you to calculate all pairwise two-time correlations within T1 and T2 and then you apply the formula of Eberly for time-dependent spectra setting the integration times to T1 and T2 and Gamma=0.

Kevin

[Kevin Fischer]

Unfortunately, you're getting beyond my expertise a bit (so I would recommend taking my advice here as a potentially incorrect starting point)! My understanding is that yes, you can approximate the system with an effective time-independent Hamiltonian when you see periodic dynamics, but the dimension of that effectively Hamiltonian is likely larger than the dimension of the original system.

Kevin