Control of atomic transition rates via laser-light shaping

A modular systematic analysis of the feasibility of modifying atomic transition rates by tailoring the electromagnetic field of an external coherent light source is presented. The formalism considers both the center of mass and internal degrees of freedom of the atom, and all properties of the field: frequency, angular spectrum, and polarization. General features of recoil effects for internal forbidden transitions are discussed. A comparative analysis of different structured light sources is explicitly worked out. It includes spherical waves, Gaussian beams, Laguerre-Gaussian beams, and propagation invariant beams with closed analytical expressions. It is shown that increments in the order of magnitude of the transition rates for Gaussian and Laguerre-Gaussian beams, with respect to those obtained in the paraxial limit, require waists of the order of the wavelength, while propagation invariant modes may considerably enhance transition rates under more favorable conditions. For transitions that can be naturally described as modifications of the atomic angular momentum, this enhancement is maximal (within propagation invariant beams) for Bessel modes, Mathieu modes can be used to entangle the internal and center-of-mass involved states, and Weber beams suppress this kind of transition unless they have a significant component of odd modes. However, if a recoil effect of the transition with an adequate symmetry is allowed, the global transition rate (center of mass and internal motion) can also be enhanced using Weber modes. The global analysis presented reinforces the idea that a better control of the transitions between internal atomic states requires both a proper control of the available states of the atomic center of mass, and shaping of the background electromagnetic field.

Figure 1

(a) Illustrative basic integrals ${\mathcal{H}}_{l,m}^{{\mathrm{w}}_{\perp }}$ for scalar Gauss modes; and (b) illustrative basic integrals ${\mathcal{H}}_{l,m}^{{\mathrm{w}}_{\perp }}$ for vector TE and TM Gauss modes. They are plotted as a function of their waist ${\mathrm{w}}_{\perp }$ measured in terms of the inverse wave number $a=\omega {\mathrm{w}}_{\perp }/c$.

Figure 2

Basic $k_{\perp }$ integrals, $h_{l,m}^{{\mathrm{w}}_{\perp }}$ for (a) scalar propagation invariant modes, (b) transverse electric and transverse magnetic propagation invariant modes as a function of the transverse wave number ${\kappa }_{\perp }$ measured in units of $\omega /c$. The waist of the Gaussian factor was taken as $10c/\omega .$

Figure 3

Absolute value of the angular spectrum factor $|{\tilde{\mathrm{\Theta }}}_{m^{\prime }}^{(\text{Mathieu}:m)}|={\mathrm{\Theta }}_{m{m}^{\prime }}$ for Mathieu modes as a function of the continuous Mathieu parameter $q=f{\kappa }_{\perp }/2$ with $f$ the focal distance that defines the elliptic coordinates, and ${\kappa }_{\perp }$ the transverse wave number.

Figure 4

Absolute value of the angular spectrum factor $|{\tilde{\mathrm{\Theta }}}_{m_{\ell }}^{(\text{Weber}:\alpha )}|={\mathrm{\Theta }}_{m_{\ell }}^{(\text{Weber}:\alpha )}$ (a) for even Weber modes, and (c) for odd Weber modes as a function of the continuous parameter $\alpha$. For all modes, ${\mathrm{\Theta }}_{m_{\ell }}^{(\text{Weber}:\alpha )}={\mathrm{\Theta }}_{-m_{\ell }}^{(\text{Weber}:\alpha )}$.