# Rabi oscillations and energy conservation

Lets consider 2 level atom with with energy of the levels $E_a$ and $E_b$. The light frequency is $\omega$ and $\frac{E_b - E_a}{\hbar} = \omega_{ab} \ne \omega$. I.e. $\delta = \omega - \omega_{ab} \ne 0$

The initial state is $\left|\psi\right>\mid_{t=0} = \left|a\right>\left|1\right>$ (atom is in a non-excited state and light has 1 photon).

The Schrodinger equation solution assumed to be in the following form

$\left|\psi\right> = C_a(t) \left|a\right>\left|1\right> + C_b(t) \left|b\right>\left|0\right>$ (1).

Thus the initial conditions are $C_a(0) = 1, C_b(0)=0$.

The solution is the following $C_b=B\sin{\frac{\Omega_R}{2} t}$, where $\Omega_R$ is the Rabi frequency. The coefficient $B$ is determined via $\omega_R$ (Rabi frequency for $\delta = 0$) as follows $B = \frac{\omega_R}{\Omega_R}$. Thus we have a non zero probability to get the system in the state $\left|b\right>\left|0\right>$ but the energy of the state is $E_b$ that is not equal to the energy of the initial state $E_a + \hbar \omega$ as soon as $\omega \ne \omega_{ab}$.

How it can happen? I think that the error is in the solution form (1). Most probably the basis was chosen incorrectly. What basis should be used? Any other ideas?

• The Rabi Oscillation will cause a the energy level to split. See Autler–Townes effect and Mollow triplet – user_na Apr 16 '17 at 20:56
• @user_na, it seems to be an answer on my question, thank you a lot. – Ivan Apr 16 '17 at 21:33
• glad I could help. – user_na Apr 19 '17 at 12:47