Two-Level Atom Driven by Two Lasers?

[Twigg]

Hey all,

I am looking for a reference that derives the optical Bloch equations for a two-level system driven by two near-detuned monochromatic radiation sources. Specifically, I am looking to substantiate a result I derived by following the same procedure as for a two-level atom driven by a single radiation source: $$\frac{d\vec{R}}{dt} = \vec{R} \times \vec{W}$$ where $$\vec{W} = [\Omega_1 + \Omega_2 \mathrm{cos}(\delta_2 - \delta_1),\Omega_{2}\mathrm{sin}(\delta_2 - \delta_1),\hbar \delta_1]^{\mathrm{T}}$$ in which $$\hbar \Omega_n = e \langle \vec{x} \cdot \vec{E_{n}} \rangle$$ and ##\delta_n## is the detuning of the n-th laser and ##\vec{E_n}## is the amplitude and polarization of the beam.

Edit: I don't need the spontaneous emission contribution, because I'm putting this in a Monte Carlo simulation where I handle spontaneous emission as a random process for each atom.

Thanks in advance!

 

[eljose79]

Hi there,

The optical Bloch equations for a two-level system driven by two near-detuned monochromatic radiation sources can be derived using the density matrix formalism. This approach takes into account the coherence between the two levels and is often used for systems with multiple driving fields.

The resulting equations are similar to the single driving field case, but with additional terms that account for the interaction between the two driving fields. These equations are given by:

$$\frac{d}{dt}\rho_{11} = -i\Omega_{1}\rho_{12} + i\Omega_{2}^*\rho_{21} + \Gamma_{11}\rho_{11}$$

$$\frac{d}{dt}\rho_{12} = -i(\Delta_{1} + \Delta_{2})\rho_{12} + i\Omega_{1}(\rho_{11} - \rho_{22}) + i\Omega_{2}\rho_{22} + \Gamma_{12}\rho_{12}$$

$$\frac{d}{dt}\rho_{22} = i\Omega_{1}^*\rho_{12} - i\Omega_{2}\rho_{21} + \Gamma_{22}\rho_{22}$$

where ##\Delta_{1}## and ##\Delta_{2}## are the detunings of the two lasers, and ##\Gamma_{ij}## are the relaxation rates.

These equations can be solved numerically or analytically to obtain the time evolution of the density matrix elements, which can then be used to calculate the expectation values of observables such as the population of each level.

I hope this helps with your research. Good luck!