How Do You Derive the Lagrangian for the Jaynes-Cummings Model?

[lxhrk9]

Homework Statement

I am trying to write down the path integral for the Jaynes-Cummings Model which involves obtaining the Lagrangian.

Homework Equations

$$\hat{H}_{\text{JC}} = \hbar \nu \hat{a}^{\dagger}\hat{a} +\hbar \omega \frac{\hat{\sigma}_z}{2} +\frac{\hbar \Omega}{2} \left(\hat{a}\hat{\sigma}_+ +\hat{a}^{\dagger}\hat{\sigma}_-\right)$$

The Attempt at a Solution

To get the Lagrangian from the Hamiltonian is it reasonable to write the creation and annihilation operators in terms of x and p, solve
$$\dot{x}=\frac{\partial H(x,p)}{\partial p}$$
for p and plug this into
$$L(x,\dot{x};t)=\dot{x}p−H(x,p;t)$$?

 

[mark726]

Thank you for your question. The Jaynes-Cummings Model is a well-known quantum mechanical model that describes the interaction between a two-level atom and a quantized electromagnetic field. In order to obtain the Lagrangian for this model, we first need to rewrite the Hamiltonian in terms of the canonical variables, x and p.

As you have correctly mentioned, the Hamiltonian can be written as:

\hat{H}_{\text{JC}} = \hbar \nu \hat{a}^{\dagger}\hat{a} +\hbar \omega \frac{\hat{\sigma}_z}{2} +\frac{\hbar \Omega}{2} \left(\hat{a}\hat{\sigma}_+ +\hat{a}^{\dagger}\hat{\sigma}_-\right)

To rewrite this in terms of x and p, we need to express the creation and annihilation operators in terms of these variables. This can be done by using the standard relations:

\hat{a}=\sqrt{\frac{m\omega}{2\hbar}}(x+ip)
\hat{a}^{\dagger}=\sqrt{\frac{m\omega}{2\hbar}}(x-ip)

Substituting these expressions into the Hamiltonian, we get:

\hat{H}_{\text{JC}} = \hbar \nu \left(\frac{m\omega}{2\hbar}(x^2+p^2)+\frac{1}{2}\right) +\hbar \omega \frac{\hat{\sigma}_z}{2} +\frac{\hbar \Omega}{2} \left(\sqrt{\frac{m\omega}{2\hbar}}(x-ip)\hat{\sigma}_+ +\sqrt{\frac{m\omega}{2\hbar}}(x+ip)\hat{\sigma}_-\right)

Now, we can use the standard definition of the Lagrangian:

L(x,\dot{x};t)=\dot{x}p-H(x,p;t)

To obtain the Lagrangian for the Jaynes-Cummings Model, we need to solve the equation:

\dot{x}=\frac{\partial H(x,p)}{\partial p}

This can be done by differentiating the Hamiltonian with respect to p, which gives us:

\dot{x}=\hbar \nu \frac{