# How to numerically solve a laser driving semi-classical two-level system using Floquet formalism?

Consider the semi-classical laser driving two-level atom, where the laser is treated classically and the atom is treated quantum mechanically. The effect of laser on the atom is a dipole coupling: $$i\,\dot\Psi(t)=[-\frac{\omega_0}{2}\sigma_z+\Omega(t)\cos(\omega t)\sigma_x]\Psi(t)$$ where $\Psi(t)$ is the wavefunction, $\omega_0$ is the two level frequency, $\omega$ is the driving frequency, $\Omega(t)$ is the coupling strength. $\sigma_x,\sigma_z$ are the Pauli matrixes. In matrix form this can be written as:

$$i \left( \begin{array}{c} \dot C_1(t) \\ \dot C_2(t) \\ \end{array} \right)=\left( \begin{array}{cc} -\frac{\omega _0}{2} & \Omega(t)\cos(\omega t)\\ \Omega(t)\cos(\omega t) & \frac{\omega _0}{2} \\ \end{array} \right)\cdot \left( \begin{array}{c} C_1(t) \\ C_2(t) \\ \end{array} \right)$$ where $C_1(t)$ and $C_2(t)$ are the state amplitude for the two states.

If the coupling $\Omega(t)$ is weak and $\omega_0\approx\omega$ then this can be solved using the Rotating-Wave Approximation. I'm intrested in the region where $\Omega$ not small compared to $\omega_0$ or $\omega$, and $\Omega(t)$ has a shape $$\Omega(t)= \left\{ \begin{array}{ll} \Omega_0\sin^2(\pi t/\tau) & 0\leq t<\tau\\[1em] 0 & \tau \leq t \end{array} \right.$$ where $\tau>>\frac{2\pi}{\omega}$. If the coupling $\Omega(t)$ is time independent, then we can use the standard Floquet theory to solve the equation. But if $\Omega(t)$ is time-dependent, how do I use the Floquet formalism to solve this problem?

• @BrandonEnright thanks, could you help me to transfer to CS? – xslittlegrass May 31 '13 at 3:37
• I flagged it for a moderator to look at. Only they can perform the migration. – Brandon Enright May 31 '13 at 3:38
• I've looked at this post a couple times, and my main question is: how will Floquet theory help you solve this equation numerically? I'm vaguely familiar with the theory; a colleague used it to develop a method for sensitivity analysis of periodic systems. Let's say that you find the time-dependent change of coordinates for your system such that you now have a linear ODE with constant coefficients. Will the solution you obtain from the transformed ODE be substantially more accurate than solving original ODE numerically? (That is, will the payoff be worth the effort?) – Geoff Oxberry Jun 1 '13 at 8:49
• @GeoffOxberry thanks for interested in the question. I think Floquet theory may not help in solving this equation numerically in accuracy, performance etc. . However, it will be very help to understand the physics of the system if we can solve using floquet formalism. For example, if the coupling $\Omega(t)$ is constant in time, the floquet theory indicates that the time-dependent equation can be transformed into a infinite set of time-independent equations, diagonalization of the infinite floquet matrix will give the floquet states and the floquet energies. – xslittlegrass Jun 1 '13 at 16:32
• @GeoffOxberry Although mathematically the floquet method is equivalent or even harder than other numerical method, it gives useful information other method may not capable of. For instance, one can easily see how the initial population distributes among the different floquet states and how the phase change in floquet states give rise to the time evolution. – xslittlegrass Jun 1 '13 at 16:33

You should probably take a look at

"Single molecule counting statistics for systems with periodic driving" J. Chem. Phys. 139, 164120 (2013) http://dx.doi.org/10.1063/1.4826634

The authors there use a truncation of the Floquet matrix to look at semiclassical two-level systems. Not sure if it covers precisely the case you are looking at, but it might be a good starting point.

You have two coupled differential equations with complex variables. You can use differential equation solvers here. There are two ways to handle such problems numerically:

1. Define suitable function with two inputs- a complex array of size 2 and time; and return the RHS of the ODE system. Pass this function to a suitable ODE solver that can handle complex numbers. For an example in Python using scipy take a look at the code at the bottom of this page: http://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.ode.html

2. Instead of complex array of size 2 use real array of size 4 and explicitly calculate the RHS of the ODE as 4 real equations. Then pass the function to the solver. Again an example in Python is at: http://www.scipy.org/Cookbook/Zombie_Apocalypse_ODEINT

Note that in both cases the RHS can explicitly depend on time. You can use matlab or any other ODE solver for this problem.

• Thanks, but I'm asking how to solve use Floquet theory, which take the advantage of the quasi-periodicity of the driving term. – xslittlegrass May 31 '13 at 13:54
• since the question was initially asked on physics.stackexchange.com i thought you might not know this stuff. what i described can simulate arbitrary time-dependent hamiltonian. i don't know if something special can be done if the time-dependence is quasiperiodic. – Rajeev Jun 1 '13 at 6:07
• Thanks for the idea, but I think a lot of solver can solve this kind of equation. I want to solve the equation specificly in floquet formalism because I'm interested in thinking about the problem in the dressed states picture. Even it's quasi-periodic, I think one can still use the Floquet formalism, there are adiabatic and non-adiabatic Floquet theory to deal with that. But I don't know how to implement that in detail. – xslittlegrass Jun 1 '13 at 6:28