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A quantum mechanical two-level system driven by a constant sinusoidal external potential is very useful in varies areas of physics. Although the widely used rotating-wave approximation (RWA) is very successful in treating weak coupling and near resonance cases, sometimes an analytical solution beyond the RWA is desired. Are there any special cases (for example large detuning, very strong driving, etc.) where one can get the analytical solutions beyond the RWA?

In mathematics, this is to say that solve the following equation analytically for $C_1$ and $C_2$:

\begin{align} i\dot{C}_1(t)&=\Omega\cos(\omega t)e^{-i\omega_0t}C_2(t)\\ i\dot{C}_2(t)&=\Omega\cos(\omega t)e^{i\omega_0t}C_1(t) \end{align} where $C_1(t)$ and $C_2(t)$ are the two level state amplitude, $\Omega$ is the coupling strength, $\omega_0$ is the two level frequency difference, and $\omega$ is the driving frequency. $\omega$, $\omega_0$, $\Omega$ are constant and $C_1$ and $C_2$ are time dependent quantities.

Any suggestions or related literatures are appreciated.

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    $\begingroup$ If I'm not mistaken Cohen-Tannoudji vol. 2 solves this exactly. $\endgroup$
    – ErickShock
    Commented Apr 13, 2019 at 20:19
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    $\begingroup$ I can see this is a very old post. In any case said equation can be solved by "going to the interaction picture". I can post a solution in case you are still interested. $\endgroup$
    – lcv
    Commented Sep 13, 2019 at 12:53

2 Answers 2

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This system can be solved exactly by Floquet theory. The Floquet theory deals with periodically driven quantum systems, and this is exactly the prototypical example where a solution can be obtained.

Floquet theory in a nutshell

Consider the time-dependent Schrodinger equation with periodic time-dependent Hamiltonian $H(t)$ (such that $H(t+nT)=H(t)$, where $n$ is an integer and $T$ the period): $$ i\hbar \partial_t \psi_{\alpha}(t) = H(t) \psi_{\alpha}(t). \;\;\;\;\;\;\;\;\; (1) $$ In analogy with Bloch theory of particles in periodic potentials, we can assume that the eigenmode $\psi_{\alpha}(t)$ can be written as $$ \psi_{\alpha}(t) = e^{-i\varepsilon_{\alpha}t/\hbar} u_{\alpha}(t) = \sum_{m=-\infty}^{+\infty} e^{-i(\varepsilon_{\alpha}/\hbar + m\omega)t} u_{\alpha}^{(m)} \;\;\;\;\;\;\;\;\; (2) $$ where $u_{\alpha}(t+nT) = u_{\alpha}(t)$ is a time periodic function with period T, and hence it can be expanded in a Fourier series with standard frequency $\omega = 2\pi/T$ and weights $u_{\alpha}^{(m)}$. Substituting (2) into (1), you get the important equation $$ \sum_n (H_{mn} - n\hbar\omega \delta_{mn}) u_{\alpha}^{(n)} = \varepsilon_{\alpha}u_{\alpha}^{(m)}; \;\;\;\;\;\; H_{mn}\equiv \frac{1}{T}\int_0^T dt H(t) e^{i(m-n)\omega t}. \;\;\;\;\;\; (3) $$ This can be recast into an eigenvalue problem by organizing the Fourier components into a vector: for example, in the two level system ($\alpha=1,2$), the vector $\vec{u}$ is: $$ \vec{u} = ( \dots , u^{(-1)}_1, \,u^{(-1)}_2 , u^{(0)}_1, \,u^{(0)}_2 , u^{(1)}_1, \,u^{(1)}_2 , \dots) $$ and the matrix we want to diagonalize is an infinite matrix made by $2\times 2$ blocks, each block being labeled by a pair of integers $(m,n)$.

Driven two level system

Let's now consider the following time periodic Hamiltonian that describes a two level system with a very specific periodic drive: $$ H(t) = \frac{\varepsilon}{2} \, \hat{\sigma}_z + \frac{V}{2} \left( \cos(\omega t) \hat{\sigma}_x + \sin(\omega t) \hat{\sigma}_y \right), $$ where $\hat{\sigma}_a$ is the $a$-th Pauli matrix. Notice that this is not the only possible choice for the drive, but this is the only one for which an analytic solution can be provided (as far as I know). The physical system we have in mind here is a spin-$1/2$ in a magnetic field along $z$ and a circularly polarized magnetic field (with constant modulus) in the $xy$ plane. The matrix to be diagonalized here is the following: \begin{equation} \frac{1}{2} \left( \begin{array}{c|cc|cc|cc|c} \ddots & & & & & & & \\ \hline & \varepsilon+2\hbar\omega & 0 & 0 & 0 & 0 & 0 & \\ & 0 & -\varepsilon+2\hbar\omega & V & 0 & 0 & 0 & \\ \hline & 0 & V & \varepsilon & 0 & 0 & 0 & \\ & 0 & 0 & 0 & -\varepsilon & V & 0 & \\ \hline & 0 & 0 & 0 & V & \varepsilon - 2\hbar\omega & 0 & \\ & 0 & 0 & 0 & 0 & 0 & -\varepsilon - 2\hbar\omega & \\ \hline & & & & & & & \ddots \end{array} \right), \end{equation} where each box is a $2\times 2$ block labeled by $(m,n)$. You can see immediately that this matrix is block diagonal (just remove the lines and look :) ), so you can easily find the eigenvalues: $$ \varepsilon_{\pm, n} = \frac{\hbar \omega \pm \hbar \omega_R}{2} - n\omega, \;\;\;\;\;\;\;\;\; (\hbar\omega_R)^2 \equiv (\hbar\omega - \varepsilon)^2 + V^2 $$ and the corresponding eigenvectors ($\left|\uparrow\right\rangle = (1,0)^T \;\; \left|\downarrow\right\rangle = (0,1)^T$): $$ u_{\pm}^m = \frac{ V \delta_{m,0} \left|\downarrow\right\rangle + (\varepsilon - \hbar\omega \pm \hbar \omega_R) \delta_{m,+1} \left|\uparrow\right\rangle }{\sqrt{V^2 + (\varepsilon - \hbar\omega \pm \hbar \omega_R)^2}}. $$

From these solutions you can get important dynamical properties of the system, e.g. the transition probability to go from $\left|\downarrow\right\rangle$ at $t=0$ to $\left|\uparrow\right\rangle$ at time $t$. I will be very sketchy here: the idea is to write down the wave function as a superposition of the two eigenmodes $\psi(t) = a \psi_+(t) + b \psi_-(t)$, find the coefficients $a$ and $b$ by the condition $\left\langle\downarrow \right| \psi(0) \rangle = 1$ and finally computing $$ P(t) \equiv |\left\langle \uparrow \right| \psi(t) \rangle|^2 = \frac{V^2}{\omega_R^2} \sin^2{(\frac{\omega_R t}{2})} $$

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You could use the Floquet theory to go beyond the RWA. See the paper by Shirley[J. H. Shirley, Phys. Rev. B 138, 974 (1965)]. If the amplitude of the driving field isn't too strong, you could use a simpler perturbative method such as the averaging method to second-order. Look here at this manuscript: Rabi oscillations in two-level systems beyond the rotating-wave approximation

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    $\begingroup$ Try to provide answers which can be used even if the links within it were to go down. Perhaps take a summary of what you think are the key points to your link and add that to your post. $\endgroup$ Commented Mar 16, 2017 at 18:32

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