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m Moved the definition of Delta to *before* its first use :) |
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=i\hbar\left(\frac{i}{\hbar}UH_0\right)U^\dagger+U(H_0+H_I)U^\dagger
=UH_IU^\dagger \\
&=-e^{i\omega_gt|\text{g}\rangle\langle\text{g}| + i\omega_et|\text{e}\rangle\langle\text{e}|}\left(
\hbar\left(\Omega e^{-i\omega_Lt}+\tilde{\Omega}e^{i\omega_Lt}\right)|\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}|\right)
e^{-i\omega_gt|\text{g}\rangle\langle\text{g}| - i\omega_et|\text{e}\rangle\langle\text{e}|}\\
&=-e^{i\,0\,t|\text{g}\rangle\langle\text{g}| + i\omega_0t|\text{e}\rangle\langle\text{e}|}\left(
\hbar\left(\Omega e^{-i\omega_Lt}+\tilde{\Omega}e^{i\omega_Lt}\right)|\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}|\right)
e^{-i\,0\,t|\text{g}\rangle\langle\text{g}| - i\omega_0t|\text{e}\rangle\langle\text{e}|}\\
&=-e^{i\omega_0t|\text{e}\rangle\langle\text{e}|}\left(
\hbar\left(\Omega e^{-i\omega_Lt}+\tilde{\Omega}e^{i\omega_Lt}\right)|\text{e}\rangle\langle\text{g}|
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