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I am having some difficulties understanding the "adiabatic elimination" in the context of atomic physics. In particular, consider a three-level system with states labeled by $|g_1\rangle$, $|g_2\rangle$ and $|e\rangle$ coupled to a classical electromagnetic field which stimulates the transitions $|g_1\rangle \leftrightarrow |e\rangle$ and $|g_2\rangle \leftrightarrow |e\rangle$. After some work, we can write the Hamiltonian in the rotating wave approximation and in a suitable frame of reference where it is time independent as $$ H = \left( \begin{array}{ccc} \delta\omega_1 - \Delta & 0 & \Omega_1 \\ 0 & \delta\omega_2 - \Delta & \Omega_2 \\ \Omega_1 & \Omega_2 & -\Delta \end{array} \right), $$ where $\delta\omega_i$ are the field detunings, $\Omega_i$ the Rabi couplings and $\Delta = (\delta\omega_1+\delta\omega_2)/2$. The basis that I have used is $\{|g_1\rangle, |g_2\rangle, |e\rangle\}$. Now this leads to a Schrodinger equation for the state $\psi_{g_1}|g_1\rangle+\psi_{g_2}|g_2\rangle + \psi_e|e\rangle$ that, for what concerns the $|e\rangle$ component of the state, reads $$ i\partial_t \psi_{e} = -\Delta \psi_e + \Omega_1 \psi_{g_1} + \Omega_2 \psi_{g_2}. $$

So far so good, but now here's my problem: the typical assumption to obtain an effective two-level dynamics from here is to set $\partial_t\psi_e=0$, which is called adiabatic elimination and apparently it holds under the assumption $\delta \omega_1 \approx \delta\omega_2$ (or similarly $|\delta\omega_i-\Delta| \ll \Delta$). However I totally don't understand why we can set $\partial_t\psi_e=0$. I understand that we are interested in slow dynamics and we want to get rid of fast oscillations; but how does this lead to $\partial_t\psi_e=0$? Can anyone please explain this subtle point to me?

Qmechanic
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Matteo
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1 Answers1

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This is well explained in Chapter 10 ("Three states") of Bruce Shore's review on "Coherent Manipulations of Atoms Using Laser Light", which you can find on the web archive here (I got a Bad Gateway error trying to access directly from the source site).

In section 10.3.1 "Adiabatic elimination", he explains that, in the limit where the pump laser detuning $\delta \omega_1$ (to transition from $|g_1\rangle$ to $|e\rangle$ in your labelling) is large, when $\delta \omega_1 \gg \Omega_1, \Omega_2$, the time derivative $\delta_t\psi_e$ varies rapidly, as you say. He says

We are not concerned with such rapid variations, and we average them out by taking an average over many cycles. The average of the derivative vanishes, and we can solve the resulting equation for the average amplitude $\overline{C_2}$

where $C_2$ is the probability amplitude for the state $|e\rangle$ in your labelling, for a general state

$|\psi\rangle = C_1 |g_1\rangle + C_2 |e\rangle + C_3 |g_3\rangle$.

He then shows that as a result, $C_2$ drops out of the equations of motion for $C_1$ and $C_3$ i.e it has been adiabatically eliminated.

I struggled a lot with adiabatic evolution/elimination until I found this text. I would thoroughly recommend reading beyond just this chapter. It also helped clarify the rotating-wave approximation for me better than any other review, textbook or paper I have found.

Ben Roberts
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