# Two-level system Hamiltonian from electric-dipole approx

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$$_{\mathrm{XIII}}$$).