Hey all,
I am trying to create a bath at finite temperature that can be coupled to an external spin, and develop the interaction between the bath and the system using TDVP because of his efficiency.
Currently, I am trying to thermalize the bath under the purification method, and afterward I want to couple it to a two level system. So what I need to do is to create a thermal state for my bath, and then to represent it as a MPS in order to couple it to the spin system and develop their interaction + self interaction of the bath in real time. I will try to write the equations in case I wasn’t clear enough.
Let me begin by creating the thermal state for the bath:
\rho=e^{-\beta H }=|\psi_{sis'}\rangle \langle \psi_{s,s'}|
Now, the s sites developed in imaginary time under my Hamiltonian, but the s’ only through the entanglement and in fact they don’t have any physically meaning.
So I want to take \rho ->| \psi _s\rangle and trace out all the ancilla sites.
And after that I want to define H'=H+H_{int}(H is exactly the same as before), and |\psi \rangle=|\psi_s \rangle \otimes |S\rangle
and then to evolve in unitary using the TDVP:
|\psi(t) \rangle = e^{-iH't} |\psi \rangle
My problem is that either I will find a way to create a thermal MPS, or to time evolve the density matrix under the self Hamiltonian+interaction term because this Hamiltonian is quite complicated.
I hope this was clear, if any further clarification needed let me know.
I basically thought that maybe I can convert the MPO to MPs using the “convert” function, and then I will be able to use the tdvp function of ITensors. Is It possible ? is the H will act only on the real site in that method ? H is a MPO object.
Hi @tsegev, thanks for explaining in more detail. You are going to need to think through your question some more, though, because some of the ideas here are not possible within the formalism of quantum mechanics.
One point is that in general, e^{-\beta H} is not a “pure” density matrix, so there is no such state \ket{\psi} such that e^{-\beta H} = \ket{\psi} \bra{\psi}, with the one exception of when \beta \rightarrow \infty (in which case \ket{\psi} would be the ground state). That’s ok though, because let’s proceed by saying you have obtained \rho = e^{-\beta H} as an MPO.
Then next you are saying you would like to trace out the ancilla sites. This is a reasonable thing to want to do in some cases, but what is the goal of doing it here? For one thing, if you trace out the ancilla sites, the resulting object will be a mixed state density matrix, not a pure state \ket{\psi_s} so the rest of your question would have to be revisited where for the time evolution part at the end you’d be time evolving a mixed state coupled to your spin, not a pure state.
So could you please clarify a bit more of what you’d like to do? Would you be ok with time evolving a mixed state MPO at the end of your procedure or is there some other reason the state can be a pure state \ket{\psi} that I might have missed?
The first part is to create a thermal state some H model. This is the part where I wrote you before.
The reason that I tend to trace out the Ancilla’s is because I think that in the second a large Hilbert decreasing the time of calculations. This is only a preparation for the main physical process I want.
The second part is the couple it to a two level system and calculate the rate of decoherence between the environment and that system where the two level system will be: |\psi \rangle= \frac{1}{\sqrt2} (|0\rangle +|1\rangle)
and then to evolve in unitary time.
The reason that I think I need to trace out the Ancilla part is because I don’t want to present the system as an MPO (I don’t want to have an Ancilla part to it) and I don’t know of any way to couple an MPO to a MPS and then to evolve them both together.
I’m not very get the question points. However, I think the simple way to study the problem may be using the Lindblad master equation or Redfiled master equation for case of Gaussian environments. If the system-bath coupling too stronger, you can apply the HEOM, PIMC, and t-DMRG, and some other non-perturbative methods.
In second relates to the question, you can introduce fictious modes with annihilator \hat{b}, which is a counterpart of operator \hat{a}. Then performing the thermal transformation that lead to from vacuum state to the thermal vacuum state. If you this relates to your question, you can refer “thermal field dynamics”.
Hi @tsegev,
Like Meng, I’m also not quite sure I understand the procedure you are proposing. But I think you’ll need to step back from thinking about performance i.e. whether tracing various kinds of sites could make your calculations run faster, and instead start with whether such steps are even physically correct to do. As you know, the purpose of introducing ancilla sites isn’t mainly about making code go faster or slower, but is actually necessary for getting the correct thermal (finite-temperature) physics.
Do you have senior colleauge or advisor who can help you evaluate your plan for your calculation and whether your formalism is correct? Mainly the purpose of this forum is to help people to use the ITensor software, so it’s difficult for us to answer very broad and general physics questions, such as about the correct formalisms for studying open quantum systems.