Dynamical model of coherent circularly polarized optical pulse interactions with two-level quantum systems

We propose and develop a method for theoretical description of circularly (elliptically) polarized optical pulse resonant coherent interactions with two-level atoms. The method is based on the time-evolution equations of a two-level quantum system in the presence of a time-dependent dipole perturbation for electric dipole transitions between states with total angular-momentum projection difference $(\Delta J_z=\pm 1)$ excited by a circularly polarized electromagnetic field [Feynman et al., J. Appl. Phys. 28, 49 (1957)]. The adopted real-vector representation approach allows for coupling with the vectorial Maxwell’s equations for the optical wave propagation and thus the resulting Maxwell pseudospin equations can be numerically solved in the time domain without any approximations. The model permits a more exact study of the ultrafast coherent pulse propagation effects taking into account the vector nature of the electromagnetic field and hence the polarization state of the optical excitation. We demonstrate self-induced transparency effects and formation of polarized solitons. The model represents a qualitative extension of the well-known optical Maxwell-Bloch equations valid for linearly polarized light and a tool for studying coherent quantum control mechanisms.

Figure 1

Energy-level diagram of a two-level quantum system with $\Delta J_z=\pm 1$ dipole-allowed transitions and energy separation $\hslash {\omega }_0$. States are labeled by pairs of total angular momentum and its projection along the propagation $z$ axis. (a) The resonant medium modeled by an ensemble of two-level systems is assumed to be initially absorbing (all atoms in their ground states). The population is driven into the upper excited level ∣2⟩ by left-circularly-polarized optical field $\left({\sigma }^{-}\right)$ and returned back to its initial state ∣1⟩ by right-circularly-polarized optical field $\left({\sigma }^{+}\right)$. (b) The system is amplifying (all atoms in the excited state). The population is driven from the initial excited level ∣1⟩ to the final ground state ∣2⟩ by right-circularly-polarized optical field $\left({\sigma }^{+}\right)$ and returned back to the excited state by left-circularly-polarized optical field $\left({\sigma }^{-}\right)$.

Figure 2

Time evolution of the $E_x$ component (a) and the $E_y$ component (c) of left-circularly-polarized pulse ${\sigma }^{-}(+,-)$ with a unit amplitude $E_0=1\mathrm{V}{\mathrm{m}}^{-1}$ sampled at the left boundary (solid line) and at the right boundary (dashed line) of the simulation domain. The pulse with duration $T_p\sim 30\mathrm{fs}$ is a cosine function given by Eq. (3.1) (a) and a sine function given by Eq. (3.2) (c) modulated by a Gaussian envelope with carrier frequency corresponding to a wavelength $\lambda =1.5\mu \mathrm{m}$ at resonance with the atomic transition. Fourier spectra of the time traces of the (b) $E_x$ component and (d) $E_y$ component in (a) and (c), respectively, at the input facet (solid line) and the output facet (dashed line) clearly showing absorption of the pulse energy at the resonant transition frequency for both $E$ field components.

Figure 3

Absorption coefficient for $E_x$ component (a) and $E_y$ component (b) of the electric field calculated from the real part of Eq. (3.22) using the FDTD solution of the Maxwell pseudospin system (dashed line) compared with the theoretical absorption coefficient as calculated from the steady-state solution of the Maxwell-Bloch system with linear polarization from Eq. (3.23) (solid line). Excellent agreement is found for both $E$ field components.

Figure 4

Absorption coefficient for $E_x$ component (a) and $E_y$ component (b) of a right-circularly-polarized optical field ${\sigma }^{+}(+,+)$ compared with the theoretically calculated absorption coefficient for linear polarization showing that the pulse does not interact with the two-level system.

Figure 5

Time evolution of the $E_x$ component (a) and the $E_y$ component (c) of a right-circularly-polarized pulse ${\sigma }^{+}(+,+)$ launched in a gain medium (initially pumped into the excited state). The corresponding Fourier spectra (b) and (d) exhibit amplification due to the emission of the absorbed energy by the carriers.

Figure 6

Gain coefficient calculated by FDTD method for the $E_x$ component (a) and the $E_y$ component (b) of a right-circularly-polarized pulse ${\sigma }^{+}(+,+)$ compared with the gain coefficient from the linear polarization theory. Excellent agreement between the theory and the FDTD result is achieved.

Figure 7

Gain coefficient calculated by FDTD method for the $E_x$ component (a) and the $E_y$ component (b) of a left-circularly-polarized pulse ${\sigma }^{-}(+,-)$ plotted together with the gain coefficient from the linear polarization theory showing failure of deexcitation of the two-level system by left-circularly-polarized light.

Figure 8

Time evolution of a $2\pi$ SIT left-circularly-polarized pulse ${\sigma }^{-}(+,-)$ propagating in absorbing medium (the initial population profile $S_{30}=-1$ shows the boundaries of the absorbing medium). The normalized electric field components $E_x$ and $E_y$ are plotted together with the population inversion profile $S_3$ at the simulation times $t=200$, 267, 334, and $400\mathrm{fs}$. The pulse travels undistorted through the two-level medium.

Figure 9

Left-circularly-polarized $2\pi$ SIT pulse ${\sigma }^{-}(+,-)$ with a hyperbolic secant envelope at the simulation time $t=200\mathrm{fs}$. The normalized $E$ field components are plotted together with the population inversion. The polarized SIT pulse completely inverts locally the two-level medium and returns the population back to the ground state within one Rabi flop.

Figure 10

Right-circularly-polarized SIT $\pi$ pulse ${\sigma }^{+}(+,+)$ completely inverts the medium returning the population back to the ground state. The normalized $E$ field components $\left(E_{x,y}\right)$ and the population $\left(S_3\right)$ are given at the simulation time $t=267\mathrm{fs}$.

Figure 11

Time evolution of a right-circularly-polarized SIT $\pi$ pulse ${\sigma }^{+}(+,+)$. The normalized $E$ field components $\left(E_{x,y}\right)$ and the population $\left(S_3\right)$ are given at the simulation time $t=120,240,360,480\mathrm{fs}$.