Add introductory tutorial on Jaynes-Cummings-Hubbard Model #130
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Vanshaj0429
changed the title
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Hi @Ericgig, I have opened this PR on possibility of adding a tutorial on Jaynes-Cummings-Hubbard model. Would love to have your feedback on it. |
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Hi @Ericgig! This is my first contribution to the QuTiP tutorials, and I’ve added a new notebook titled The notebook:
It looks like the CI failed because of dead links in some unrelated files:
Since those aren’t touched by this PR, I believe the failure isn't caused by my changes. |
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| # Introduction to the Jaynes-Cummings-Hubbard Model: Three-Site System | ||
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| Authors: [Your Name] |
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Feel free to put your name here!
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| $$i\hbar\frac{d|\psi(t)\rangle}{dt} = H|\psi(t)\rangle$$ | ||
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| In QuTiP, we use the `mesolve` function to numerically solve this equation. For a closed quantum system without dissipation, the dynamics will be purely coherent, showing quantum oscillations as photons hop between cavities and interact with atoms. |
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Since this is a closed system, it would be a bit more usual to use sesolve. (mesolve will just delegate to sesolve in this case)
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| # Calculate time evolution using the master equation solver | ||
| # For a closed system without dissipation, this solves the Schrödinger equation | ||
| result = mesolve(H, psi0, tlist, [], ops['cavity_n'] + ops['atom_e']) |
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| result = mesolve(H, psi0, tlist, [], ops['cavity_n'] + ops['atom_e']) | |
| result = mesolve(H, psi0, tlist, [], e_ops=(ops['cavity_n'] + ops['atom_e'])) |
(or sesolve)
e_ops will be a keyword-only parameter from qutip 5.3.
| return np.mean(delta_n_values), np.mean(alpha_values) | ||
| ``` | ||
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| Now let's calculate these order parameters across a range of hopping strengths to see how they change as we approach and cross the phase transition: |
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I am a bit confused here since I do not really see anything in the plots I get. In fact,
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Here's what's happening as per my understanding:
For a finite-size system (especially a small 3-site system), the ground state preserves certain symmetries (this is my educated guess) that cause the expectation value of the field operator (⟨a⟩) to be exactly zero, even across the phase transition. The photon number fluctuations (blue circles) still increase as expected, showing a precursor to the phase transition.
In larger systems (let's say extremely large), ⟨a⟩ would become non-zero in the superfluid phase, serving as a true order parameter. But in our small system, we need to look at other quantities (like photon number fluctuations) to see signatures of the transition.
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| ### Theoretical Background on Photon Propagation | ||
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| In the JCHM, photons don't simply move from one cavity to the next in a classical way. Instead, they show quantum mechanical wave-like behavior, with interference effects and probability amplitudes spreading across the lattice. The propagation pattern depends on both the hopping strength J and the cavity-atom coupling strength g. |
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I am sorry, but this plot confuses me. Is it possible that the axes labels are switched?
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No axes are fine, there was an orientation issue which made it hard to interpret. I have fixed that now, and you could see the below screenshot once I make the commit:
This is heatmap now represents the correct dynamics as show below:
Hi @pmenczel! Thanks for reviewing. I have been in talks with Eric to make some contribution as part of my GSoC application. I am a recent Master's graduate in physics from Cardiff University. There I worked on a project that focused on simulating features of Jaynes Cummings model using QuTiP. I also wanted to explore the Jaynes Cummings Hubbard model, but couldn't do it at the time due to time constraints. This tutorial is somewhat an extension of my previous work trying to simulate more rich models. https://github.com/Vanshaj0429/cavity-qed-simulations If you have any more questions or feedback. I am happy to answer :) |
Summary
This tutorial introduces the Jaynes-Cummings-Hubbard Model on a three-site cavity array, showcasing how to build and simulate light–matter interactions using QuTiP. It walks through Hamiltonian construction, ground state analysis, time evolution, and phase transition signatures, offering a compact, hands-on guide to exploring quantum dynamics in coupled cavity QED systems.
The notebook is written in Jupytext Markdown format and adheres to the qutip-tutorials contribution guidelines. All code cells execute successfully and pass
pytest --nbmake.Tutorial Highlights
Key Concepts Covered:
Construction of the full JCHM Hamiltonian for three coupled atom-cavity sites
Analysis of the ground state properties and atomic excitation
Time evolution of localized photon excitations using mesolve
Exploration of phase transition signatures using:
Effects of:
Tools & Techniques
tensor,qeye,destroy, andsigmazeigenstatesqutip.mesolvematplotlibFile Location
tutorials-v5/miscellaneous/JCHM-tutorial.mdAdditional Notes
Let me know if: