Rotating-wave approximation: Difference between revisions

Revision as of 15:40, 18 February 2020

Revision as of 06:52, 2 July 2020

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: <math>U = e^{iH_0t/\hbar} = e^{i \omega_0 t |\text{e}\rangle \langle\text{e}|} = |\text{g}\rangle \langle\text{g}| +e^{i \omega_0 t} |\text{e}\rangle \langle\text{e}|</math>,

where the last step can be seen to follow e.g. from a [[Taylor series]] expansion with the fact that <math>|\text{g}\rangle\langle\text{g}|+|\text{e}\rangle\langle\text{e}|=1</math>, and due to the orthogonality of the states <math>|\text{g}\rangle</math> and <math>|\text{e}\rangle</math>. weThe substitution for <math>H_0</math> in the second step being different from the definition given in the previous section can be justified either by shifting the overall energy levels such that <math>|\text{g}\rangle</math> has energy <math>0</math> and <math>|\text{e}\rangle</math> has energy <math>\hbar \omega_0</math>, or by noting that a multiplication by an overall phase (<math>e^{i \omega_0 t/2}</math> in this case) on a unitary operator does not affect the underlying physics. We now have

: <math>\begin{align}