Confining Stationary Light: Dirac Dynamics and Klein Tunneling

We discuss the properties of 1D stationary pulses of light in an atomic ensemble with electromagnetically induced transparency in the limit of tight spatial confinement. When the size of the wave packet becomes comparable or smaller than the absorption length of the medium, it must be described by a two-component vector which obeys the one-dimensional two-component Dirac equation with an effective mass $m^{*}$ and effective speed of light $c^{*}$. Then a fundamental lower limit to the spatial width in an external potential arises from Klein tunneling and is given by the effective Compton length ${\lambda }_C=\hslash /(m^{*}{c}^{*})$. Since $c^{*}$ and $m^{*}$ can be externally controlled and can be made small, it is possible to observe effects of the relativistic dispersion for rather low energies or correspondingly on macroscopic length scales.

Figure 1

Stationary light scheme: The interaction of double $\Lambda$ atoms driven by two counterpropagating control fields of (equal) Rabi frequency $\Omega$ and opposite circular polarization with two counterpropagating probe fields $E_{\pm }$ of corresponding polarization generates a quasistationary pattern of the probe fields.

Figure 2

Top: Diffusive expansion of an initial Gaussian stationary light pulse with width $L_p=2.5{\lambda }_C$. Bottom: The same for an initial width of $L_p=0.05{\lambda }_C$. Results are obtained from numerical solutions of the full Maxwell-Bloch equations which agree very well with solutions of the Schrödinger equation (5) (top plot) and the Dirac equation (9) (bottom plot). The parameters are $\gamma /\Delta =0.01$, ${\Omega }_{\pm }/\Delta =0.2$, $\cos \theta =0.9975$, and hence ${\lambda }_C/L_{\mathrm{abs}}=199.5$.

Figure 3

Evolution of an initial Gaussian stationary light pulse with initial width $L_p=0.05{\lambda }_c$ in a square-well potential (the extension of which is indicated by dashed red lines) with $a=0.1{\lambda }_C$ and depth $U_0=1.875m^{*}{c}^{*2}$ (top), and $U_0=3.125m^{*}{c}^{*2}$ (bottom), respectively. Besides initial losses due to mismatch of the wave functions [see insets at the top, $I(t)$ is the integrated intensity], a fast decay due to Klein tunneling is apparent in the second case.

Figure 4

Left: Center of mass of emerging left- and right-moving wave packets obtained from solution of Maxwell-Bloch equations of initial Gaussian wave packet of width $\Delta z=10L_{\mathrm{abs}}\ll {\lambda }_C=199.98L_{\mathrm{abs}}$ (solid blue line) and from solution (17) of the Dirac equation (dashed red line). At $t=0$ a flip of the relative phase of the drive fields was assumed corresponding to ${\mathcal{E}}_{+}(0)=i{\mathcal{E}}_{-}(0)$. Parameters are as in Fig. 2. Right: Fourier-spectrum of $⟨z(t)⟩$ showing the pronounced peak at $2m^{*}{c}^{*2}/\hslash$.