Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  The requirement of a relatively weak coupling for the application of the rotating wave approximation to obtain the Jaynes-Cummings model

+ 6 like - 0 dislike
1461 views

In order to obtain the Jaynes-Cummings Hamiltonian, the RWA is applied to the Rabi Hamiltonian: $$H=\frac{1}{2}\hbar\omega_0 \sigma_z+\hbar\omega \hat{a}^{\dagger}\hat{a}+\hbar g(\sigma_{+}+\sigma_{-})(\hat{a}+\hat{a}^{\dagger}),$$ under two conditions: the near resonance $\omega_0\approx\omega$, and the relatively weak coupling strength $g\ll \text{min}\{\omega_0,\omega\}$. While the requirement for the first condition is reasonable (the terms $\sigma_{+}\hat{a}^{\dagger}$ and $\sigma_{-}\hat{a}$ become rapidly oscillating for $|\omega_0-\omega|\ll \omega_0+\omega$, as seen in the interaction picture w.r.t. the first two terms in $H$, where they acquire phase factors $e^{\pm i(\omega_0+\omega)t}$), the condition for the weak coupling is not that evident and usually is not explained in the introductory quantum optics textbooks. So, why is it needed?


This post imported from StackExchange Physics at 2017-03-22 18:45 (UTC), posted by SE-user Andyk

asked Mar 4, 2017 in Theoretical Physics by Andyk (30 points) [ revision history ]
edited Mar 22, 2017 by Dilaton
This is a good question. I don't think I've ever seen a really quantitative analysis of the conditions under which the RWA work, much less an analysis of the error in various limits. This issue is of particular interest to me because of this experiment.

This post imported from StackExchange Physics at 2017-03-22 18:45 (UTC), posted by SE-user DanielSank
@DanielSank possibly linked to the non-inertial nature of the rotating frame?

This post imported from StackExchange Physics at 2017-03-22 18:45 (UTC), posted by SE-user ZeroTheHero
@ZeroTheHero The rotating wave approximation appears in first order differential equations, such as Hamilton's equations of motion in classical physics or the analogous Heisenberg equations in quantum mechanics. Neither of these cases have inertia, so I'm not sure what you mean by non-inertial nature.

This post imported from StackExchange Physics at 2017-03-22 18:45 (UTC), posted by SE-user DanielSank
Could you give a source where the condition $g \ll min\{\omega_0,\omega\}$ is clearly stated ? Indeed to my knowledge, which might be wrong, in order to move from the Rabi Hamiltonian to Jaynes-Cummings Hamiltonian, you first go to Interaction Picture with $\hat{H}_{int} = \frac{1}{2}\hbar\omega_0\sigma_z + \hbar\omega\hat{a}^{\dagger}\hat{a}$, then neglect the terms $\propto e^{\pm i (\omega_0 + \omega)t}$, then finally move back to the Schödinger Picture. I seems to my that the weak coupling-strength does not play any role here, or maybe I missed something ...

This post imported from StackExchange Physics at 2017-03-22 18:45 (UTC), posted by SE-user mhham
Okay it seems that my comment was wrong, look at my answer below !

This post imported from StackExchange Physics at 2017-03-22 18:45 (UTC), posted by SE-user mhham

1 Answer

+ 3 like - 0 dislike

I will not go into much detail here, but rather give you the link to this answer.

In order to sum it up very quickly let us simply state that the RWA which gives rise to the Jaynes-Cummings Hamiltonian is an on-resonance perturbative theory, where we neglect the fast rotating terms in the Rabi Hamiltonian when written in the interaction picture.

In the answer, a simple model was given where an atom is classically driven by a field. The coupling constant is thus proportional to the driving field, and it is stated that :

It is essential to emphasize that, as the applied field increases, this approximation becomes even less reliable and it is just the leading order of a perturbation series in a near-resonance regime.

This is a direct analogue of the $g \ll min\{ω_0,ω\}$ condition.

Hence one could say that the Rabi and Jaynes-Cummings Hamiltonian describe the same physics as soon as both conditions (near-resonance and weak coupling) are verified. If the coupling becomes strong (as in superconducting qubits for instance), the Jaynes-Cummings Hamiltonian no longer describes completely the physics, since higher order terms start to play a role. (cf. Bloch-Siegert shift and/or AC Stark shift).

An interesting paper on this topic : A modern review of the two-level approximation by Marco Frasca.

Edit : Also, a very elegant way to look at these light-atom interaction problems, is through the dressed-atom formalism (Atom-Photon Interactions - Chapter 6 The Dressed Atom Approach by Claude Cohen-Tannoudji , or any introductory ressource that builds the dressed-atom approach starting from the Rabi Hamiltonian and not the Jaynes-Cummings one).

This post imported from StackExchange Physics at 2017-03-22 18:45 (UTC), posted by SE-user mhham
answered Mar 13, 2017 by mhham (30 points) [ no revision ]
So, briefly speaking, the need for small $g$ can be seen when expanding the time evolution operator in interaction picture (Dyson series). For JC model the higher order terms are negelcted when $g$ is much smaller than $\omega_0$ and $\omega $ (i.e. $\mathcal{O}[g^2/(\omega_0+\omega)^2]$). And in combination with $\omega_0\approx \omega$, the counterrotating terms can be negelcted .

This post imported from StackExchange Physics at 2017-03-22 18:45 (UTC), posted by SE-user Andyk
Exactly, you got it !

This post imported from StackExchange Physics at 2017-03-22 18:45 (UTC), posted by SE-user mhham
I also insist on the fact that when $g$ is no longer small with respect to $\omega_0 , \omega$, you do not describe completely the physics (i.e Bloch-Siegert shift and/or AC Stark shift) with the JC model.

This post imported from StackExchange Physics at 2017-03-22 18:45 (UTC), posted by SE-user mhham
Experimental example : THz driven quantum wells: Coulomb interactions and Stark shifts in the ultrastrong coupling regime, Zaks et al.

This post imported from StackExchange Physics at 2017-03-22 18:45 (UTC), posted by SE-user mhham

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\varnothing$ysicsOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...