Electromagnetically Induced Transparency in Gas Cells with Antirelaxation Coating

Abstract

The features of electromagnetically induced transparency (EIT) in gas cells with antirelaxation coating of the walls are considered. It is shown that the simultaneous effect of the motion of atoms, the finite size of a cell, and the nondegeneracy of the ground state of atoms leads to a number of qualitative phenomena. In particular, even for the simplest three-level lambda scheme with a nondegenerate ground state, a significant difference in the spectra of EIT is revealed and investigated in detail depending on the relation between the frequencies of the probe and control fields, i.e., depending on whether the probe radiation is scattered in the Stokes or anti-Stokes channel. A significant spatial inhomogeneity of the excited low-frequency coherence is observed, which persists for large cells even far from their boundaries. In this case, the EIT resonances formed by atoms in different regions of a cell turn out to be differently shifted in frequency, which causes an additional inhomogeneous broadening of the observed profile. The observed effects are studied as a function of the nature of the accommodation of atoms on the surface of the antirelaxation coating.

APPENDIX A

Neglecting the collisions of atoms with each other, we can write the system of equations for the slow amplitudes of the density matrix in the Wigner representation ρ(v, r, t) in the rotating wave approximation as

$$\begin{gathered} {{{\dot {\rho }}}_{{11}}} + {\mathbf{v}} \cdot \nabla {{\rho }_{{11}}} = i(V_{c}^{*}{{\rho }_{{31}}} - {{V}_{c}}{{\rho }_{{13}}}) + {{\gamma }_{{31}}}{{\rho }_{{33}}}, \\ {{{\dot {\rho }}}_{{22}}} + {\mathbf{v}} \cdot \nabla {{\rho }_{{22}}} = i(V_{p}^{*}{{\rho }_{{32}}} - {{V}_{p}}{{\rho }_{{23}}}) + {{\gamma }_{{32}}}{{\rho }_{{33}}}, \\ {{{\dot {\rho }}}_{{12}}} + {\mathbf{v}} \cdot \nabla {{\rho }_{{12}}} = i(V_{c}^{*}{{\rho }_{{32}}} - {{V}_{p}}{{\rho }_{{13}}}) \\ \, + [i({{\Delta }_{p}} - {{\Delta }_{c}} + {\mathbf{q}} \cdot {\mathbf{v}}) - {{\gamma }_{{12}}}]{{\rho }_{{12}}}, \\ \end{gathered} $$

$$\begin{gathered} {{{\dot {\rho }}}_{{13}}} + {\mathbf{v}} \cdot \nabla {{\rho }_{{13}}} = - i(V_{c}^{*}{{\rho }_{{11}}} + V_{p}^{*}{{\rho }_{{12}}} - V_{c}^{*}{{\rho }_{{33}}}) \\ - \,[i({{\Delta }_{c}} - {{{\mathbf{k}}}_{c}} \cdot {\mathbf{v}}) + {{\gamma }_{3}}{\text{/}}2]{{\rho }_{{13}}}, \\ {{{\dot {\rho }}}_{{23}}} + {\mathbf{v}} \cdot \nabla {{\rho }_{{23}}} = - i(V_{c}^{*}{{\rho }_{{21}}} + V_{p}^{*}{{\rho }_{{22}}} - V_{p}^{*}{{\rho }_{{33}}}) \\ - \,[i({{\Delta }_{p}} - {{{\mathbf{k}}}_{p}} \cdot {\mathbf{v}}) + {{\gamma }_{3}}{\text{/}}2]{{\rho }_{{23}}}, \\ {{\rho }_{{11}}} + {{\rho }_{{22}}} + {{\rho }_{{33}}} = 1,\quad {{\rho }_{{ij}}} = \rho _{{ji}}^{*}. \\ \end{gathered} $$

(A.1)

Here we assume that the control field induces transitions only from sublevel |1〉, and the probe field, only from sublevel |2〉. We also use the following notations: V_c = d₃₁ ⋅ E_c/2$\hbar$; V_p = d₃₂ ⋅ E_p/2$\hbar$; d_eg are the dipole moments of the transitions; E_c and E_p are the slow amplitudes of the control and probe fields, respectively; γ₃ = γ₃₁ + γ₃₂; q = k_c – k_p; Δ_c = ω_c – ω₃₁; and Δ_p = ω_p – ω₃₂. For brevity we omit the arguments of the density matrix. In system (A.1), we also neglected the probabilities of spontaneous transitions between closely spaced levels |1〉 and |2〉.