Nonlinear effects in pulse propagation through Doppler-broadened closed-loop atomic media

Nonlinear effects in pulse propagation through a medium consisting of four-level double-$\Lambda$-type systems are studied theoretically. We apply three continuous-wave driving fields and a pulsed probe field such that they form a closed interaction loop. Due to the closed loop and the finite frequency width of the probe pulses, the multiphoton resonance condition cannot be fulfilled, such that a time-dependent analysis is required. By identifying the different underlying physical processes we determine the parts of the solution relevant to calculate the linear and nonlinear response of the system. We find that the system can exhibit a strong intensity-dependent refractive index with small absorption over a range of several natural linewidths. For a realistic example we include Doppler and pressure broadening and calculate the nonlinear self-phase modulation in a gas cell with sodium vapor and argon buffer gas. We find that a self-phase modulation of $\pi$ is achieved after the propagation of a few centimeters through the medium while the absorption and pulse shape distortion in the corresponding spectral range is small.

Figure 1

(Color online) The four-level atomic system with the four dipole-allowed transitions forming a closed-loop double-$\Lambda$-type scheme. Three transitions are driven by continuous-wave control fields indicated by the solid blue double arrows. The fourth transition couples to the pulsed probe field indicated by the dashed red double arrow. The coupling strengths are given by the Rabi frequencies ${\Omega }_{jk}$. The spontaneous decays with rates ${\gamma }_{jk}$ are denoted by the wiggly green lines ($j\in \left\{3,4\right\}$, $k\in \left\{1,2\right\}$).

Figure 2

(Color online) Real part (solid blue line) and imaginary part (dashed red line) of the linear susceptibility of the probe field. Due to strong control fields ${\Omega }_{42}=100\gamma$ and ${\Omega }_{31}=50\gamma$, the probe field resonance is split into four different resonances. Further, ${\Omega }_{32}={\Delta }_{31}={\Delta }_{32}={\Delta }_{42}=0$ and all spontaneous decay rates ${\gamma }_{jk}$ have been set to $\gamma$. The susceptibility is plotted in units of $3/8{\pi }^2{\lambda }_{41}^3{N}_s$.

Figure 3

(Color online) Real part (dash-dotted blue line) and imaginary part (solid red line) of the nonlinear susceptibility together with the imaginary part of the linear susceptibility (dashed red line). All figures show the resonance around ${\Delta }_{41}=-25\gamma$. The susceptibility is plotted in units of $3/8{\pi }^2{\lambda }_{41}^{3}N$, and for comparability ${\chi }^{\left(3\right)}$ has been scaled with $3/4E_{41}^2$. The parameters are ${\Delta }_{32}={\Delta }_{42}=0$, ${\Omega }_{31}=50\gamma$, ${\Omega }_{32}=34\gamma$, and ${\Omega }_{42}=100\gamma$. The probe field strength is assumed to be one-tenth of the weakest control field in all cases. The detuning ${\Delta }_{31}$ is chosen as (a) ${\Delta }_{31}=0$, (b) ${\Delta }_{31}=0.7\gamma$, (c) ${\Delta }_{31}=1.5\gamma$, and (d) ${\Delta }_{31}=1.7\gamma$. Note the different axis scales in the four subpanels.

Figure 4

(Color online) Real part (dash-dotted blue line) and imaginary part (solid red line) of the nonlinear susceptibility together with the real part (blue dotted line) and the imaginary part of the linear susceptibility (dashed red line) at the resonance around ${\Delta }_{41}=-15.0\text{GHz}$. The control fields have Rabi frequencies ${\Omega }_{42}=60\text{GHz}$, ${\Omega }_{31}=30\text{GHz}$, and ${\Omega }_{32}=25\text{GHz}$, and the detunings are ${\Delta }_{31}=1.6\text{GHz}$ and ${\Delta }_{32}={\Delta }_{42}=0$. The medium parameters described in the main text correspond to sodium as the active medium with argon as a buffer gas. The four different plots show Doppler-averaged results with a Doppler linewidth of (a) below the natural linewidth, (b) 50%, (c) 90%, and (d) 100% of the full Doppler linewidth of $\delta \omega =2\pi \times 1.78\text{GHz}$. In plot (d) we also included the second derivative of the real part of the linear susceptibility (long-dashed black line).

Figure 5

(Color online) A magnification of Fig. 4d around the probe field frequency ${\Delta }_{41}^{\text{min}}$ with vanishing group velocity dispersion [see Eq. (28)] is shown. The frequency axis is shown in units of the natural decay rate $\gamma$, which for the considered sodium $D1$ transition is $2\pi \times 9.76\text{MHz}$. The parameters are the same as for Fig. 4d.