• 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.


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


(propose a free ad)

Site Statistics

204 submissions , 162 unreviewed
5,026 questions , 2,180 unanswered
5,344 answers , 22,683 comments
1,470 users with positive rep
815 active unimported users
More ...

  Quantum optical master equation

+ 2 like - 0 dislike

In Quantum Noise by Gardiner and Zoller (p. 86f) they derive and write a version of the quantum optical master equation.

$ \dot{\rho}_S (t) = -\frac{i}{\hbar} [H_\mathrm{sys} ,\rho_S] - \sum_m \frac{\pi \omega_m}{2\hbar } \bigg( \bar{N}(\omega_m) +1 \bigg) \kappa (\omega_m)^2 [\rho_S X_m^+ -X^{-}_{m} \rho_S , X] + \mathrm{ ... more\, terms ... }$

Eq. 3.6.64 (and eq. 3.6.67) do not make dimensional sense to me. The units of the derivative of the density operator should be frequency.

The units of terms in the sums are those of $\omega_m \kappa^2 X^2/\hbar $ where $\kappa^2$ was defined (on p. 45) to be the spring constant of a harmonic oscillator.

On that page they also write energy terms as $\kappa^2 X^2$. This means that the terms have dimension of frequency-squared.

Is there a mistake in the book or am I misunderstanding the meaning of some part of the equation? (the derivation of the equation is not given in the book; they write instead "one finds, after some labour".

More definitions from the book

X is a system operator and $ X = \sum_m (X_m^+ + X_m^- )$ where

$ [H_\mathrm{sys}, X_m^\pm ] = \pm \hbar \omega_m X_m^\pm$

and the total system Hamiltonian is written (eq. 3.1.5) as

$ H = H_\mathrm{sys} + \frac{1}{2} \sum_n \big[ (p_n-\kappa_n X)^2 + \omega_n^2 q_n^2 \big] $

This post imported from StackExchange Physics at 2015-11-02 22:13 (UTC), posted by SE-user HansHarhoff
asked Jun 22, 2015 in Theoretical Physics by HansHarhoff (10 points) [ no revision ]
Please do not rely on external links for content which is crucial to the question. I know it is a long equation but you should still transcribe it in full, or at the very least provide a screenshot.

This post imported from StackExchange Physics at 2015-11-02 22:13 (UTC), posted by SE-user Emilio Pisanty

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:
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).
To avoid this verification in future, please log in or register.

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

Your rights