How can I derive Eq 9.5.11 in Scully's Quantum Optics

In summary: Is it possible to do that?In summary, the equation of motion for an operator ##(a^\dagger)^ma^nO_A## in the Heisenberg picture is given by$$\frac{d}{dt}[(a^\dagger)^ma^nO_A]=-\frac{i}{\hbar}[(a^\dagger)^ma^nO_A,H_F+H_A+H_{AF}]+\langle\frac{d}{dt}[(a^\dagger)^ma^n]\rangle_RO_A$$
  • #1
cube6991
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Homework Statement
In the Scully's Quantum Optics textbook, section 9.5, an equation 9.5.11 is given to calculate the time derivative of the operator $$(a^\dagger) ^ma^nO_A$$. I try to derive this equation by myself, however, there are several problems.
Relevant Equations
Eq-9.1.1, Eq-9.3.5, Eq-9.5.5~10
Firstly, I don't know in which Picture this equation holds (if I hadn't missed some words in the previous text...). I think it may be the Heisenberg Picture. But if it is, the rest target is to prove $$\frac{i}{\hbar}[H_R+H_{FR},(a^\dagger) ^ma^nO_A]=\langle\frac{d}{dt}((a^\dagger) ^ma^n)\rangle_RO_A$$. Left side of this equation is something dependent on $\hat{b}$, but the right side is not. So how can I prove this equation?

(This is my first time to post a problem in Physics Forum, so if I did something wrong remind me please and I will correct.)
Thanks very much!
 
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  • #2
I think you have a lot of typos in the equation above it seems inconsistent. For starters, you seem to have ensemble average on the RHS, and just the pure operators on LHS. Please check and define all terms. Also, can you please present the question in a self-contained way, so that people would not need to check the book to understand what you mean?

Looking beyond this, what seems to be the problem? You have an operator, you have the Hamiltonian, you find the commutator of the former with the latter, and this is your time-derivative.

But you have already written it all that. So what is the question?
 
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  • #3
Cryo said:
I think you have a lot of typos in the equation above it seems inconsistent. For starters, you seem to have ensemble average on the RHS, and just the pure operators on LHS. Please check and define all terms. Also, can you please present the question in a self-contained way, so that people would not need to check the book to understand what you mean?

Looking beyond this, what seems to be the problem? You have an operator, you have the Hamiltonian, you find the commutator of the former with the latter, and this is your time-derivative.

But you have already written it all that. So what is the question?
Sorry for these mistakes. I will present this question in a self-contained way tomorrow or the day after tomorrow because the book is not at hand now. Thank you for your reply.
 
  • #4
The whole question is: (No typo this time I think...)
In section 9.5 of Scully's quantum optics, an atom in a damped cavity is researched. The Hamiltonian is
$$H=H_F+H_A+H_{AF}+H_R+H_{FR}$$
The subscript F represents Field, and A -> Atom, R ->reservoir and AF -> interactions between atom and field and so on.
$$H_F=\hbar\nu a^\dagger a$$
$$H_A=\frac{1}{2}\hbar\nu\sigma_z$$
$$H_{AF}=\hbar g(\sigma_+a+a^\dagger\sigma_-)$$
$$H_R=\sum_{k}\hbar\nu_kb^\dagger_k b_k$$
$$H_{FR}=\hbar\sum_k g_k(b^\dagger_ka+a^\dagger b_k)$$
where ##\sigma_z=|e\rangle\langle e|-|g\rangle\langle g|##, ##\sigma_-=|g\rangle\langle e|##,##\sigma_+=(\sigma_-)^\dagger##.
Then it says that the equation of motion for any operator of the form ##(a^\dagger)^ma^nO_A##, (where ##O_A## is an atomic operator) is given by
$$\frac{d}{dt}[(a^\dagger)^ma^nO_A]=-\frac{i}{\hbar}[(a^\dagger)^ma^nO_A,H_F+H_A+H_{AF}]+\langle\frac{d}{dt}[(a^\dagger)^ma^n]\rangle_RO_A$$
and the term ##\langle\frac{d}{dt}[(a^\dagger)^ma^n]\rangle_R##is given by an equation in the previous section:
$$\frac{d}{dt}\langle(a^\dagger)^ma^n\rangle_R=[i\nu(m-n)-\frac{\ell}{2}(m+n)]\langle(a^\dagger)^ma^n\rangle_R+\ell mn\bar{n}_{th}\langle(a^\dagger)^{m-1}a^{n-1}\rangle_R$$
However, this equation (above equation, in the previous section) is derived in Heisenberg picture using the Hamiltonian
$$H=H_0+H_1$$
$$H_0=\hbar\nu a^\dagger a+\sum_k\hbar\nu_kb^\dagger_kb_k$$
$$H_1=\hbar\sum_kg_k(b^\dagger_ka+a^\dagger b_k)$$

In current section (not the previous section), the picture is not specified. But form the form of the motion equation, I guess it is in interaction picture where ##H_0=H_R+H_{FR}##and##H_1=H_A+H_F+H_{AF}##. But I still can not derive this motion equation, and another question is that why one could apply the equation in the previous section (it is in the Heisenberg picture) to current section (in interaction picture)?

Besides, you may notice that the difference between ##\langle\frac{d}{dt}(a^\dagger)^ma^n\rangle_R## and ##\frac{d}{dt}\langle(a^\dagger)^ma^n\rangle_R##, but the original text is so.

PS. I try to find an "EDIT" button to modify my post directly, but I could not find such a button (using Chrome or Edge)...
 
  • #5
Cryo said:
I think you have a lot of typos in the equation above it seems inconsistent. For starters, you seem to have ensemble average on the RHS, and just the pure operators on LHS. Please check and define all terms. Also, can you please present the question in a self-contained way, so that people would not need to check the book to understand what you mean?

Looking beyond this, what seems to be the problem? You have an operator, you have the Hamiltonian, you find the commutator of the former with the latter, and this is your time-derivative.

But you have already written it all that. So what is the question?
Thank you. I have updated this post with more information. Now it is self-contained in some degree...
 

1. How do I know which equation is Eq 9.5.11 in Scully's Quantum Optics?

Eq 9.5.11 in Scully's Quantum Optics refers to a specific equation in the book "Quantum Optics" by Marlan O. Scully and M. Suhail Zubairy. It can be found in Chapter 9, Section 5, Equation 11.

2. What is the significance of Eq 9.5.11 in Scully's Quantum Optics?

Eq 9.5.11 in Scully's Quantum Optics is a mathematical expression that describes the dynamics of a two-level atom interacting with a radiation field. It is a fundamental equation in the field of quantum optics and is used to study the behavior of atoms and light.

3. Can I derive Eq 9.5.11 in Scully's Quantum Optics on my own?

Yes, it is possible to derive Eq 9.5.11 in Scully's Quantum Optics on your own if you have a strong understanding of quantum mechanics and mathematical techniques such as perturbation theory and the rotating wave approximation. However, it is recommended to have a thorough understanding of the concepts and equations leading up to Eq 9.5.11 before attempting to derive it.

4. Are there any resources available to help me derive Eq 9.5.11 in Scully's Quantum Optics?

Yes, there are many resources available such as textbooks, online lectures, and research papers that explain the derivation of Eq 9.5.11 in Scully's Quantum Optics. It is also helpful to consult with a mentor or professor who has experience in the field of quantum optics.

5. How can I apply Eq 9.5.11 in Scully's Quantum Optics to real-world situations?

Eq 9.5.11 in Scully's Quantum Optics can be applied to various real-world situations such as understanding the behavior of atoms in a laser, designing quantum communication systems, and studying the interaction between light and matter in quantum computing. It is a powerful tool for analyzing and predicting the behavior of quantum systems.

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