Phase-controlled pulse propagation in media with cross coupling of electric and magnetic probe field component

Light propagation is discussed in media with a cross coupling of the electric and magnetic component of an applied probe field. We derive the wave equations for a probe pulse propagating through such a medium and solve them analytically in Fourier space using the slowly varying envelope approximation. Our analysis reveals the influence of the different medium response coefficients on the propagation dynamics. We apply these results to a specific example system in which cross couplings are induced in an atomic medium by additional control fields. We show that the cross couplings render the propagation dynamics sensitive to the relative phase of the additional fields, and this phase dependence enables one to control the pulse during its propagation through the medium. Our results demonstrate that the magnetic field component of a probe beam can crucially influence the system dynamics already at experimentally accessible parameter ranges in dilute vapors.

Figure 1

(Color online) Atomic level scheme used to describe the medium. Two strong control fields with Rabi frequencies ${\Omega }_1$ and ${\Omega }_2$ prepare a coherent superposition of states $|1⟩$ and $|4⟩$. The magnetic and electric probe beam components couple to two transitions with Rabi frequencies ${\Omega }_B$ and ${\Omega }_E$, respectively. ${\Omega }_C$ describes an additional control field. A possible combination of relative state parities is denoted by the $+$ and $-$ labels.

Figure 2

(Color online) Propagation dynamics in a medium with cross coupling. The closed-loop phase ${\Phi }_0=\pi /2$ is kept constant, see inset. The absolute value (solid black lines), the real part (dashed blue lines), and the imaginary part (dotted red lines) of the probe pulse envelope are plotted for successive propagation times. Arbitrary, but consistent units are used and successive plots are vertically offset by the overall phase at the center of the pulse. The spatial phase dependence of each pulse (dashed green straight lines) is compared to the value expected from the analytical solution (thick gray straight line).

Figure 3

(Color online) Propagation dynamics in the medium while the closed-loop phase is switched from $\pi /2$ to zero and back to $\pi /2$ throughout the propagation, as shown in the inset. The absolute value (black solid lines) of the probe pulse envelope is plotted for successive propagation times. The scaling and offsets are used as in Fig. 2. At the beginning and at the end of the propagation the spatial phase dependence of each pulse (dashed green straight lines) is compared to the value expected from the analytical solution (thick gray line). During the time period of zero control phase, probe pulse envelope phase does not change and the pulse experiences gain instead, as indicated by the thick dashed red line.

Figure 4

(Color online) Real (dashed blue line) and imaginary (solid red line) part of $\beta$ against the closed-loop phase ${\Phi }_0$.