After making the electric-dipole approx., I can express the interaction of a monochromatic field with angular frequency $\omega$ and a dipole moment ${\bf \mu(x)}$ as

$V({\bf x},t) = - {\bf \mu(x)} \cdot E_0\, sin(\omega t + \phi)$.

I read that in many circumstances, truncation to only a finite number of quantum states is adequate. Apparently the two state approximation results in this expression:

$H_{TLS}(t) = -\frac{1}{2} (E_2 - E_1) \sigma_z - \mu E_0\, sin(\omega t + \phi)\, \sigma_x$.

How does one arrive at this two-level approximation?


You are essentially asking for the secular approximation. The time evolution of the system is governed by the hamiltonian $$ \hat{H} = \hat{H}_0 + V(x,t),$$ where $$V(x,t) = -exE_0\frac{1}{2i}\left(e^{i\omega t}-e^{-i\omega t}\right).$$ Suppose we know the solutions of the time-dependent Schrödinger equation governed by $\hat{H}_0$ (unperturbed problem) to be $$ \psi_n(t) = \left|\left. n\right>\right. e^{-i\omega_n t},$$ where the states $\left|\left. n\right>\right.$ are eigenstates of $\hat{H}_0$, and $\hbar\omega_n$ are the corresponding eigenvalues. Then we can expand the solutions of the time-dependent problem in terms of the eigenstates of the unperturbed problem according to $$ \psi(t) = \sum_n a_n(t)\left|\left. n\right>\right. e^{-i\omega_nt}.$$ Inserting this Ansatz into the time-dependent Schrödinger equation governed by $\hat{H}$ gives a system of equations for the coefficients $a_n(t)$, namely, $$ i\hbar\partial_t a_m(t) = -\frac{eE}{2i}\sum_n a_n(t)x_{mn}\left(e^{i(\omega_{mn}+\omega)t}-e^{i(\omega_{mn}-\omega)t}\right)$$ with $\omega_{mn}=\omega_m-\omega_n$.

Now we assume that the excitation frequency $\omega$ is very close to the transition frequency $\omega_{fi}$ between an initial state $i$ and a final state $f$, i.e., $$ \omega = \omega_{fi}+\epsilon.$$ We further assume that at $t<0$ the system is in the initial state $\left|\left. i\right>\right.$, and that the excitation $V(x,t)$ is switched on abruptly at $t=0$.

The secular approximation takes in such a case only the coefficients $a_i(t)$ and $a_f(t)$ into account. This approximation leads to the two coupled equations $$ i\hbar\partial_t a_i(t) = -\frac{eE}{2i}a_f(t)x_{if}e^{+i\epsilon t}$$ $$ i\hbar\partial_t a_f(t) = +\frac{eE}{2i}a_i(t)x_{if}^\star e^{-i\epsilon t}.$$ This system of equations can be solved exactly. To this end, we define new coefficients $b_i(t)$ and $b_f(t)$ according to $$ a_i(t) = e^{+i\epsilon t/2}b_i(t)$$ $$ a_f(t) = e^{-i\epsilon t/2}b_f(t)$$ and insert them into the pair of equations above. This leads to $$ i\hbar\partial_t b_i(t) = \frac{\hbar\epsilon}{2}b_i(t) - \frac{eE}{2i}x_{if}b_f(t)$$ $$ i\hbar\partial_t b_f(t) = \frac{eE}{2i}x_{if}^\star b_i(t) - \frac{\hbar\epsilon}{2}b_f(t). $$ This pair of equations reduces to the harmonic oscillator equations $$ \partial_t^2 b_{i/f}(t) + \Omega^2b_{i/f}(t) = 0,$$ where $$ \Omega = \sqrt{\left(\frac{eE}{2\hbar}\right)^2|x_{if}|^2+\left(\frac{\epsilon}{2}\right)^2},$$ which is called the Rabi frequency.

A more elaborate discussion of the secular approximation is found, e.g, in Cohen-Tannoudji, Quantum Mechanics, Vol. II, chapter XIII (see in particular the complement C$_{\mathrm{XIII}}$).


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