Rotating-wave approximation: Difference between revisions

Revision as of 17:42, 8 December 2007

Revision as of 14:58, 13 April 2008

Line 90:

+\hbar\left(\tilde{\Omega}^*e^{-i\omega_Lt}+\Omega^*e^{i\omega_Lt}\right)|\text{g}\rangle\langle\text{e}|\right)

e^{-i\omega_0t|\text{e}\rangle\langle\text{e}|}\\

\end{align}</math>

&=-\hbar\left(\Omega e^{-i\omega_Lt}+\tilde{\Omega}e^{i\omega_Lt}\right)e^{i\omega_0t}|\text{e}\rangle\langle\text{g}|

Using a Taylor series expansion of the exponential,

<math>\begin{align}

e^{i\omega_0t|\text{e}\rangle\langle\text{e}|} = 1 + i\omega_0t|\text{e}\rangle\langle\text{e}| + \ldots

\end{align}</math>

and operating on <math>\begin{align}|\text{e}\rangle\langle\text{g}| ,

\end{align}</math>

<math>\begin{align}

e^{i\omega_0t|\text{e}\rangle\langle\text{e}|}|\text{e}\rangle\langle\text{g}| &= (1 + i\omega_0t|\text{e}\rangle\langle\text{e}| + \ldots)|\text{e}\rangle\langle\text{g}| \\

&= |\text{e}\rangle\langle\text{g}| + i\omega_0t|\text{e}\rangle\langle\text{g}| + \ldots \\

&= (1 + i\omega_0t + \ldots)|\text{e}\rangle\langle\text{g}| \\

&= e^{i\omega_0t}|\text{e}\rangle\langle\text{g}| , \\

\end{align}</math>

so the new Hamiltonian becomes

<math>\begin{align}

\bar{H}&=-\hbar\left(\Omega e^{-i\omega_Lt}+\tilde{\Omega}e^{i\omega_Lt}\right)e^{i\omega_0t}|\text{e}\rangle\langle\text{g}|

-\hbar\left(\tilde{\Omega}^*e^{-i\omega_Lt}+\Omega^*e^{i\omega_Lt}\right)|\text{g}\rangle\langle\text{e}|e^{-i\omega_0t} \\

&=-\hbar\left(\Omega e^{-i\Delta t}+\tilde{\Omega}e^{i(\omega_L+\omega_0)t}\right)|\text{e}\rangle\langle\text{g}|