Transverse Confinement of Photon Position in the Light-Atom Interaction

In an atom interferometry experiment, the output phase shift depends on the wave vector of photons and the recoil momentum in optical transitions. This Letter puts forward a hypothesis that in light-atom interaction, the atom wave function could provide a transverse confinement to photons and thus could affect the mean recoil momentum. We propose a model to analyze the photon effective wave vector in a monochromatic optical field and calculate the relative shift of $|{\overrightarrow{k}}_{\mathrm{eff}}|$ to $k$ when an atom with a 3D Gaussian wave function absorbs one photon in a Gaussian beam. This shift could lead to a systematic effect related to the spatial distribution of atoms and the transverse beam profile in high-precision experiments based on atom interferometry.

Figure 1

(a) An incident plane wave (red dashed line), $u(x,y,z)=e^{ikz}$, diffracted by a single slit. The transmission rate of the slit is $t(x,y)=\mathrm{rect}(x/a)$ and $a$ is the slit’s width. The modulus square of the angular spectrum of the diffracted light is $|A^{\prime }(k_x,k_y){|}^2=a^2\mathrm{sinc}(a{k}_x/2\pi {)}^2$. (b) Geometric illustration of the interaction between a plane-wave light field and a plane atomic matter wave (black dashed line). The atom’s wave function is ${\psi }_{\mathrm{atom}}(x,y,z)=e^{ipz/\hslash }$. There is no transverse confinement in this case. (c) Interaction between a plane-wave light and a 3D Gaussian atomic cloud (black dots) with radius ${\sigma }_a$. The atom’s transverse probability distribution is $|{\psi }_{\mathrm{atom}}(x,y){|}^2=(1/2\pi {\sigma }_a^2)e^{-(x^2+y^2)/4{\sigma }_a^2}$. The atom’s position distribution could provide a transverse confinement to the absorbed photons and lead to a projection of their transverse position states. For the absorbed photons, the transverse wave vector probability distribution becomes $|{\varphi }^{\prime }(k_x,k_y){|}^2=(1/\pi {\sigma }_a^2)e^{-{\sigma }_a^2(k_x^2+k_y^2)}$.

Figure 2

(a) $w_0=7.2$ mm, beam wavelength $\lambda =780\mathrm{nm}$, and $\delta$ as a function of the atomic cloud radius. (b) ${\sigma }_a=1.7\mathrm{mm}$, beam wavelength $\lambda =780\mathrm{nm}$, and $\delta$ as a function of the beam waist $w_0$.

Figure 3

${\sigma }_1=0.45\mathrm{mm}$, $w_0=10\mathrm{mm}$, $\lambda =780\mathrm{nm}$, and $\delta v_r/v_r$ as a function of $\eta$. (a) Reducing the transverse atomic radius to ${\sigma }_2$. (b) Blowing away the atoms located at the high-intensity region of the Gaussian beam.