Inhomogeneous mass trap for dark-state polaritons in atomic media

Abstract

The generation of a trapping potential for dark-state polaritons in a two-dimensional electromagnetically induced transparency system is theoretically studied. We show that such a trap can arise from a spatially inhomogeneous effective mass of the dark-state polariton. Because this mass inhomogeneity can be engineered by tuning the parameters of the control fields, the motion, spatial profile, and coherent behavior of bound dark-state polaritons can be tailored accordingly. Our results enable spatial controls of optical information and provide a possible route toward realizing Bose-Einstein condensation of dark-state polaritons in a trapping potential.

Figures (4)

• Figure 1: (a) Two-dimensional counter-propagating EIT system. Green dots represent the atomic medium of length $L$. Yellow arrows denote the counter-propagating biased-Gaussian control fields $\Omega_c^{F(B)}$, and blue arrows indicate the probe fields $\Omega_p^{F(B)}$. (b) Three-level $\Lambda$-type scheme, where $\Omega_p^{F(B)}$ and $\Omega_c^{F(B)}$ drive the transitions $\ket{1} \rightarrow \ket{3}$ and $\ket{2} \rightarrow \ket{3}$ with detuning $\Delta_p = \Delta_c$. Inhomogeneous mass trap potential $U_0/\hbar$ with (c) $\alpha = -0.25$ and (d) $\alpha = 0.25$. Red-solid, green-solid, and blue-solid line depict the real part of $U_0/\hbar$ for $\Delta_p=-\Gamma$, $0$, and $\Gamma$, respectively. Black-dashed line illustrates the imaginary part of $U_0/\hbar$. Other parameters are $\left( w_0, \phi, \xi\right) = \left( 1 \text{ mm}, 0, 80\right)$.
• Figure 2: (a) $\vert \psi_{00} \vert^2$ and (b) $\vert \psi_{10} \vert^2$ are the cross sections along white-dashed lines in (c) ground- and (d) first-excited-state $\vert \rho_{21} \vert^2$, respectively. Green-solid line is the steady-state $\vert \rho_{21s} \vert^2$ from OBE, red-dash-dot lines are the eigenfunction of Eq. \ref{['eq6']}, and blue-dash lines are Eq. \ref{['eq20']}. (e) Half width $\sigma$ of $\vert\psi_{00}\vert^2$. The white cross pinpoints parameters $\left( \Delta_p, w_0, \phi, \xi\right) = \left( 0, 1 \text{ mm}, 0, 80\right)$ used in (a-d). (f) Red squares, green circles, and blue triangles are $\chi$ from the numerical solution of OBE with above parameters except $\Delta_p=-0.5\Gamma$, $0$, and $0.5\Gamma$, respectively. Color solid lines are Eq. \ref{['eq22']}.
• Figure 3: (a) Coherent state oscillation of the expectation value $\left\langle z\right\rangle$ for $\Delta_p=-2\Gamma$ (red squares), $\Delta_p=-1\Gamma$ (orange circles), $\Delta_p=0\Gamma$ (green triangles), $\Delta_p=1\Gamma$ (blue-inverted triangles), and $\Delta_p=2\Gamma$ (purple diamonds). The insets depict the snapshot of $\vert \rho_{21}\left( z,y\right) \vert^2$ for $\left( \Delta_p, w_0, \phi, \xi\right) = \left( 1\Gamma, 1.5 \text{ mm}, 0, 80\right)$ at (b) $t=2\mu s$, (c) $t=42\mu s$, and (d) $t=92\mu s$.
• Figure 4: (a) $\phi$-dependent expectation value $\langle y \rangle$ at $z=0$. The red-solid line shows the numerical solution of the OBE, and the black-dashed line represents Eq. \ref{['eq23']}. (b) Im$\left[ U_0\right] /\hbar$ for $\phi = 0$ (red-solid line), $\phi = \phi_c = 0.12\pi$ (green-dashed line), and $\phi = 0.15\pi$ (blue-dashed-dotted line). (c) $\vert \rho_{21}(z,y) \vert^2$ for $\phi = 0$, (d) $\phi = 0.12\pi$, and (e) $\phi = 0.15\pi$. Other parameters are $\left( \Delta_p, w_0, \xi\right) = \left( 1\Gamma, 1.5 \text{ mm}, 200\right)$.